“If you had told me that three hours of Norwegian peace negotiations, with the Palestine Liberation Organisation thrown in, would be a Broadway hit and go to the National and the West End, I wouldn’t have put money on that,” laughs JT Rogers.

But this unpromising scenario has brought the American playwright to the National Theatre in London to attend auditions for its staging of Oslo, with a London commercial transfer already booked, before he flies back to New York for the Tony awards, where his play has seven nominations.

Oslo tells the previously unexplored story behind the peace process that led, in September 1993, to Israeli prime minister Yitzhak Rabin and PLO chairman Yasser Arafat shaking hands in Washington, with President Bill Clinton beaming between them like a marriage celebrant. This moment was brokered by Terje Rød-Larsen, the central character of the play, a Norwegian diplomat who arranged secret meetings between Israeli and Palestinian representatives.

Although the Scandinavian initiative showed the shape of a possible future two-state solution, the hopeful photo on the White House lawn now looks like science fiction, as the Israeli and Palestinian positions have widened and entrenched.

“It’s amazing how the historical memory deletes quite big details,” Rogers says. “In New York, you could literally feel the audience going: ‘Oh, the PLO! I’d forgotten them!’ So you wonder if, in a few years, audiences at a play will be going: ‘al-Qaida!’ or ‘Isis!’”

If contemporary American playwrights ever formed a cabinet, Rogers would be a shoo-in for secretary of state. All his major plays deal with foreign affairs: The Overwhelming (2006) dramatises the Rwandan genocide and Blood and Gifts (2010) explores the wars in Afghanistan. He has also written plays set in Spain, Germany and now Norway.

“It was never a plan,” says Rogers. “But, in hindsight, it makes sense. I’m the child of 60s Californian liberals and my father has just retired as a professor of political science. So international relations were sort of the talk at the dinner table. And we lived overseas, in south-east Asia, for a few years when I was a child. But it’s more that these are the subjects I’m interested in, rather than: wouldn’t this make a great play?”

JT Rogers … ‘I’m starstruck by public policy officials’ Photograph: Santiago Felipe/Getty

He has found, though, that geographical distance is liberating: “I see that actors are often released when they put on a disguise or a character far away from their own background. And I think the further away that people or environments are from me, the more I find it accessible. If you asked me to write a play about a guy in his mid-40s who lives in Brooklyn, I’d worry about doing something phony.”

Oslo – like Blood and Gifts and The Overwhelming – discusses serious political ideas within the form of a thriller, and this is Rogers’s own diplomatic negotiation with the audience: “I think the structures of genre get short shrift from critics. But there’s something in thrillers, melodramas and even screwball comedy – depending on the project – that give you an underpinning that is wonderfully useful for holding a script together. You can take structural elements from a genre but then do something completely different with them. I think there’s something very satisfying and compelling about experiencing a well-told tale, regardless of what the subject matter or ideas are.”

What diplomats might call the Rogers doctrine – using popular forms to put unpopular material on stage – may be rooted in his path to drama; unlike most younger American dramatists, he didn’t do an MA in playwriting, but started as an actor. “A lot of what’s coming out of the university courses is suspect to me. It often feels like plays for other people who write plays, as opposed to Sam and Steve who have just had a baby and they want a night out. We have to entertain people first and then we can do all the other things.”

As preparation for writing Oslo, Rogers chose, counter-intuitively, to read Noël Coward comedies: “I was looking for that lightness of touch and pace across the stage, although with very different material.”

Oslo reminded me of Stoppard’s Rosencrantz and Guildenstern Are Dead, in that we see a familiar narrative from a surprising angle. “I hadn’t thought of that. But that’s right: I’ll take any Stoppardian parallel. I’d wanted to write about the Middle East, and just hadn’t been able to get close enough to the Israeli or Palestinian side and then there was suddenly, in Clintonian terms, this third way.”

As he wrote, he realised that the Norwegian perspective has the advantage of being a neutral route through an issue to which audiences are likely to bring strong prejudices in either direction.

Rogers interviewed Larsen at length, but spoke to only a few of the other participants: “I stalked the characters, through memoirs and TV interviews. But the lines on stage are all mine; there’s no verbatim. My rule, though, was that no one expresses views that they didn’t hold.”

Playwrights of bio-dramas about living people dread accusations of falsification or betrayal. Has that happened? “One person reportedly didn’t like it but had the wisdom or good manners not to say it to me. As far as I know, no one has been offended or pissed off by the play. I think that’s partly because I tend only to put on stage people I either like or admire. And, in the cases where I changed people, they seem to have accepted that it was to make the complexity of the arguments more theatrical. In some way, to honour the people you have to violate them.”

Seems like sci-fi now … the historic handshake between Rabin and Arafat, recreated in Oslo. Photograph: T Charles Erickson Photography

Rogers jokes that “I’m not really star-struck by actors but I am star-struck by public policy officials”, and Oslo has filled his autograph book. At one Broadway performance, the entire 1,000-seat theatre was booked for the United Nations. They didn’t react to a joke about Arafat that generally works well because, Rogers suspects, “they all either knew him or were terrified of being seen laughing at him”. Bill Clinton, a keen theatre-goer, has presumably been? “Not yet. But very close confidants of both Bill and Hillary have told me that they are working fervently to get them to the play.”

Until Oslo, Rogers’s work has tended to premiere, unusually for an American writer, in London: “You have a strong tradition of political or state-of-the-nation plays, and we in America just don’t. Until recently, a lot of my colleagues thought I was English because of the type of plays I write and because my whole career was really started here. People even asked me why Oslo wasn’t opening here first.”

From his double perspective, Rogers throws out a thought that would keep a university drama course going for a year: “Bad American plays are invariably inward-looking and solipsistic, and the bad British plays are usually incredibly stilted, didactic and lecturing. Somewhere in the middle lies great theatre. As a theatre-goer and -maker, I just tend more towards the English style.”

Arthur Miller and Stephen Sondheim both concluded, late in their careers, that they preferred having work done in London because New York theatre seemed to be driven by economic rather than artistic considerations. But Rogers is more optimistic: “We do seem to be having a renaissance. I spent years railing against the state of American culture but recently I’ve wondered if maybe I should stop yelling. Four new American plays about politics are up for the Tony this year.”

At the Tony awards, there are likely to be a lot of anti-Trump jokes and speeches, and the current presidency seems an obvious subject for Rogers. “I haven’t been asked formally. But, informally, I’m the guy people ask to write about politics. There are stories that would be better poems, plays, movies, documentaries or whatever. With Trump, you’d need to find out what the play would be. At the moment, Saturday Night Live skits and Twitter are the best forum for it. Maybe down the line there’ll be a bit of distance. But you have to find the story; you can’t just set out to write a play about politics or it would be as boring as hell.”

• Oslo opens at the National Theatre, London SE1, on 5 September before transferring to the Harold Pinter theatre, London SW1, from 30 September.