Question:
What prompted this current email is the likelihood of Jesus speaking in Greek in any private conversation. In this case I refer to his conversation with Nicodemus in John chapter 3.  I refer to the use of the Greek word anothen as the meaning of "again/anew" and "from above." In NT Greek the word can mean either and does so in different contexts.  A viable interpretation of this passage depends on a deliberate word play on the part of Jesus. Some commentaries indicate that the Greek NT word play was a deliberate choice of Jesus. Since the Aramaic apparently does not contain the same double meaning, it is possible that Jesus spoke the conversation in Greek. The only other possibility is that John, the writer, used generous liberties in his editing of the narrative.

Answer:
I too have reflected on this question about Greek or Aramaic in the sequence with Jesus and Nicodemus.  It is not a matter of great import to me, more of a curiosity.  I am cautious in my general approach to interpreting implications of specific dialogue or word choice.

The overall theme and style of the particular book gives us some clues.  For the modern western analytical point of view, with its factual, analytical, "forensic" purposes, it is lamentable that the eastern oral-relational worldview in general does not focus on the specific details.

But those details are "missing" because that is not what is in focus in oral-relational cultures, also called concrete-relational.  (I address the character of oral-relational worldviews in several articles.)  Rather, in the biblical texts, the story as a whole is in focus, as is common in oral-relational societies.  Maybe what we are asking is not wha the writer of the passage was concerned about.

Oral Story Format
In this way the Gospels, just as Paul's letters, and broader Koine and even classical Greek "literature," represent oral culture styles of communication, rather than our modern concept of objective history.  An important aspect of this is that they were written to be read aloud.  So the effect is what is heard.

Literacy in the Roman Empire is thought to be about 10% at the time of Jesus.  Writing is a support to oral-relational communication.  What is in focus is not individual bits if information, details and facts, but relational factors.  Thus we see the Hebrew texts focus on covenants, relationships.  The same pattern continues in the Gospel teachings about the Kingdom of God, the Rule of God in our lives.  Stories are central to communication in oral-relational cultures.

The orientation of such Story-form worldviews in general especially differs in form and intent from the western forensic reporting approach.  Thus the modern concept of history, focusing on details and facts, is a new phenomenon, developing recently in western rationalist culture.  This literate, informational orientation skews our approach to these ancient testimonies that come to us from a different thought-context.

I am keen to glean every possible clue we can about the details, but our modern worldview can impose requirements on the ancient text that don't fit the cultural expectations and patterns of the text and its time.  We ahve to meet the story on its own terms, not ours.

Method
Before I look at the specifics of the encounter with Nicodemus, step back from the situation a minute and look at some practical factors.  Envision how this story might have gotten written, decades after the death of Jesus.  Think about their day-to-day situation.  Did they have tape recorders to capture word-by-word exchanges? Was that important in that culture? Did the Twelve walk around with tablets, writing down everything?

Here's another practical question.  When Nicodemus came to Jesus, was John standing nearby taking Hansard court-reporting notes to record the exchange and dialogue word-for-word, with a burning desire to get it down "on the record" for posterity?

So what should be our approach to this and the whole Gospel text?  Was his goal to reproduce a specific dialogue, or does the story have deeper meaning in the context of the whole Gospel as a unit?  What should we demand of this scene?

Modern analytical approaches focus on individual words, while most world cultures focus on the story or scene as a whole for the meaning.  From their extensive and detailed analysis and comparisons of languages all over the world, modern linguists have long told us likewise that meaning does not reside in individual words, but in the thought-phrases and syntax structures into which we put them.  The underlying shared worldview of the audience is the basis for the meaning of those structures that give individual words their meaning and life.

Context
Let's step back and let the flow of the story pull us into the Gospel, rather than pulling this one scene out of the Gospel as an independent item out of its context.  The intent of this event in John 3 has to arise from and be consistent with the overall apologetic or evangelistic purpose of this whole skillfully and tightly woven book.

This scene sits as a component alongside the stories about individuals being pushed out of the synagogue because they accept or proclaim Jesus as Messiah, like the story of the man healed by Jesus on the Sabbath.  A requirement that this be only a literal reporting account of a word-for-word dialogue that John actually overheard is an imposed requirement.

At any rate, the cultural format and literary characteristics of the Gospels may account for the word choice we see.  This phrase "born fr om above" that intrigues us and focuses the story from this point is an example of the word plays characteristic of the Fourth Gospel, and seem to reflect considered literary style for making the points of the Gospel.

That is the goal and motivation of this word play and the whole scene.  Greek was the language medium.  Story was the communication format.

Role of Jesus
This Nicodemus passage has so long been taken out of context that it stands on its own for most people today who know the story.  Could even 1 in 10 church members tell you how this story relates to the rest of the Gospel?

But this passage takes its meaning from the thrust of the whole Gospel, and Chapter 3 is one scene in a tight-knit and moving story of strife in the Jewish community splitting over the question of the Messiah.

Modern readers subconsciously draw upon their scientific analytical thinking that focuses on collection of individual facts.  Modern thinking expects this to be a news report, a compilation of facts of a reflection of facts in the background.  But it is a component of the whole narrative dealing with a relational crisis in the Jewish community.  This Gospel is organized around the theme of the growing tension between those who accept Jesus and those who don't.

Jesus is contrasted with Moses, and the question is "Who are the true Jews?" (See my study on this theme, The Unity of Jesus And God In The Fourth Gospel.)  Treating this event and other scenes from the gospel out of context as news reports misses the very point of the integrated Gospel as a whole.

The unified message I see when I read this Gospel is not an informational report for the historical record of later centuries.  This was an immediate and powerful message of proclamation and comfort to the believers, to the immediate audience who would hear this story read out.

Unified Theme
The language of the Nicodemus scene fits the pattern of the overall themes of this Gospel.  The dialogue appears tailored to meet that need and purpose.  Thus we allow the text to retain its internal integrity.  It is not wrenched from its context to meet external modern standards of format or vocabulary.

Thus the dialogue fills its role within the unity of the Fourth Gospel.  The Greek language medium is the focus, the dialogue is chosen to represent the broader goal of affirming to the hearers of this sweeping story the radical different concept of God's Rule, led by the Spirit, not by the Sword.  The theme of conflict and tension in the synagogue between followers and rejecters of the messiah underlies this scene, as every other.

Moses and Messiah
The choice of the Better Life, the Higher Life the Deeper Life is required.  Later John will portray Jesus as the Better Moses, the Better Messenger from God, the Better Deliverer, the More Authoritative Authority.  To follow Moses is to follow Jesus.  To reject Jesus is to reject Moses.  Who are the True Jews.  This seems to be the underlying question throughout this Gospel.

The vocabulary and dialogue John chose fill this need beautifully, even more powerful in hearing than in reading.  Be born from above in the Spirit, be born anew in the Spirit, a new way to live and a new focus of Life, as God's covenant kingdom enters a new stage.  The Messiah has come! It is a new beginning.  A fresh start! A Spirit start! A Spirit Life! This rich, oral play on words would hit the audience powerfully!

Let's think about the original purpose and target for this Gospel.  If this was a Greek-speaking community (attested to by the text and the good Greek and historical support), the theme and point of the Gospel would have to be made in Greek.  As the original query points out, this word play is Greek in character.  The Greek language and community would be the context, the starting point and motivation for this scene in the Fourth Gospel dramatic narrative.

Communication with that target community is the goal.  Probably the believing community, a church in one region, and the broader social community around them.  This is the focus and determining factor in the story format and word choice, for the story-teller to provide the desired impact and message of the Gospel.  This is not a wooden, objective news report, but one integral component in a powerful, moving proclamation of Life.

All Gospels
The same would hold true in each other Gospel, in regard to the broader intentions we can see and patterns we can discern in the overall structure, style, language patterns and so forth.  Put another way, it does not appear from the general form and style of the Fourth Gospel that the writer's (or editor's) purpose was to report on details of events, but to present powerful scenarios to interpret the belief in Jesus as the Messiah to those Jews who did not accept him as such and who were apparently reacting against the Jewish followers of Jesus at the time.

Much of the Fourth Gospel revolves around the role or acceptance of the Jesus-followers in the synagogue.  The event with Nicodemus is constructed consistent with this theme in the Gospel as a whole.  This powerful pun is the point of the story, it seems to me, not what specific language they were speaking.  John wrote in Greek, so this is the word-pair that clues the hearer-reader to the meaning of this event, and indeed the encounters and messages of the whole book.

This background notwithstanding, at the actual practical level outside this story, it does seem that the language of Rome (Greek) would be an odd choice between these two Jewish individuals.  In the literal social setting, though Greek would be used by the ruling elite, which would include Nicodemus as an elder and a member of the Sanhedrin, he would surely speak Aramaic also.

Aramaic was the Jewish language, though shared with other peoples, so in a day-to-day exchange I'd expect Aramaic.  It might be so because one is a Judean and the other a Galilean.  But I don't think that was the focus of John's story here.

All this is speculation and stands aloof from the point and character of the dramatic scene itself.  The word pun used in Jesus' point about the birth from above (anothen) is specifically a device in the Greek.  This could be a literary device or motif of the Gospeler, and this is the simplest accounting for the scene.  Story-form is not an objective news reporting format like we expect in modern informational terms.

Social
We realize that the story is written in Greek, and the portrayal brings out the multiple levels of meaning this Gospel is so good at.  We see exactly the same thing when we watch a movie or read a story in English about a historical situation set in another time, culture and locale with a different language.  The point of the story, the events, the characters and the themes the story intends to portray are developed in English.  In cultural patterns and phraseology or idion modern English speakers will understand.  I see this pattern in the Gospels.

The focus is: who would hear/read this story?  What would be the impact and effect of this event on the audience?  Encouragement for the despondent followers who are persecuted?  A more sympathetic attitude from the Jew who has not yet come to see Jesus as the Messiah?  It seems this would be primary.

Proclaiming, not Reporting
This is a question of the integrity of the Gospel.  What was its message, and why was it being written?  It is an Evangel, a Good News story, a proclamation.  As with all oral-relational cultures, the Story IS the Message.  The details are included only as necessary to move the story.

It seems that this is not a question of liberties in editing, but how the writer could most effectively portray Jesus as the Messiah to his audience in their cultural worldview setting.  The purpose of this story-event in John 3 would not be to provide a Hansard record of the dialogue, but to proclaim to his audience that Jesus is the Messiah, come as the messenger and message of the Rule of God.  The story as a unit fits hand in glove with this theme and the overall flow and structure of the Gospel.  That's what the story is about.

Also related:
Christians Started with a Greek Old Testament
Culture and Shared Experiences
Eye Learning or Ear Learning?
Greek and Aramaic Among 1st Century Jews
Jesus and the Hebrew Language
The Languages Jesus Used
Life, Resurrection and Judgement: the Hope of the Believing Community (John 5:19-30)
Orality in Christian Mission
Orality, Literacy and the Bible
Socialization and Self-Identity
Storytelling for Learning and Teaching
The Unity of Jesus And God In The Fourth Gospel
That Abominable Greek?
Time or Character, The Ages or A Time Sequence in aionios: How Words "Mean" in Greek and English

Also View Presentations
Orality and Post-Literate Culture
Oral and Literate – Contrast of Oral and Literate Perspectives
What is Worldview?

Related on the Internet:
Concepts of Orality in Oral Literature
Orality And Storytelling - Matters Of The Heart
Traditional Storytelling Around the World

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