While a century or so apart, both writers were masters of dissecting human foibles in a way that lent a pathos to their characters even as some of them looked increasingly ridiculous.
The result of Byrne's interest in Chekhov was Uncle Varick, which relocates Chekhov's 19th-century tale of love and life to the rural heart of north-east Scotland in the thick of the 1960s, - a time alleged in far-off London to be swinging.
Uncle Varick was first seen 10 years ago at the Royal Lyceum Theatre in Edinburgh in a towering production that featured Brian Cox in the title role in an all too rare stage role on home turf. A decade on, and the assistant director of that production, Michael Emans, is taking the helm for a major touring revival of the play, produced by his increasingly ambitious Rapture Theatre.
"I was very pleased indeed," Byrne says with an almost boyish glee about the revival, which previewed in Lanark last night prior to its official opening in East Kilbride tomorrow. "It hasn't been done by anybody since the first production, and I'm sure it was because Brian Cox was so powerful in it that it put people off."
For all the success of Uncle Varick on stage, Byrne was far from enamoured with the radio production that followed.
"It was done on Radio 3," Byrne recalls, "and they could have had the same cast, as they were working on it in the theatre, but they recast it, and I couldn't listen to it. It wasn't right. The voices were wrong, and all sounded the same. It was absolute stupidity and wrong-headedness. It was ludicrous."
While Byrne remains aghast at the radio version of Uncle Varick, he is clearly a fan of its source, even as he reinvented it.
"I don't know how Chekhov does it," Byrne says, "how he moves you and makes you laugh. It's a total mystery to me. I saw a production of Uncle Vanya on television many years ago with Laurence Olivier, and it was wonderful, just so human and lovely, but I've also seen some of the dullest versions of Chekhov that make a total arse of it by being too reverential.
"I had to think of these people as new characters invented by me, and not be reverential. There's no point in genuflecting. If you're too respectful you do the play a disservice. But when I first saw it I had a laugh myself. You cannae be too funny with Chekhov at all. The funnier you are, the darker you are with Chekhov. At the end, you're left with something that's totally about human beings, and you see the full story of life. You really do."
For Emans, fully taking the reins of Uncle Varick after cutting his directorial teeth on the original production is something he clearly relishes.
"I love the play," he says. "I'm guided by plays that fire me up, and I love Chekhov and I love John's writing, so to bring them both together in this way - and to make Chekhov accessible in the way John has - seems like a wonderful contradiction, so it's been great trying to match Byrne and Chekhov. The play works on so many levels, and I can't remember when a Chekhov play last toured."
Under Emans's guidance, Rapture have carved something of a niche for themselves in terms of reviving contemporary Scottish plays. In the last two years alone, Rapture have toured new productions of Gregory Burke's debut play, Gagarin Way, Hector MacMillan's 1970s classic, The Sash and Mike Cullen's neglected 1990s drama, The Collection. There was also a hugely popular tour of Mamma Mia! writer Catherine Johnson's earlier play, Shang-a-Lang.
Emans formed Rapture in 2000 after training as a director in London.
He'd grown up in East Kilbride, then a regular stop-off point for the era's touring companies such as 7:84, Wildcat and Borderline which first inspired him. His own company, named after David Hare's play, The Secret Rapture, initially produced three or four shows a year, and their output remains prolific.
"We're now only doing one or two pieces a year," says Emans, "but we've built up relationships with the King's Theatres in Glasgow and Edinburgh as well as other venues, so we're doing those productions to a greater degree, with a lot of time spent touring. I'm very keen for people to be able to get access to good quality theatre in places where some theatre companies might not normally go to, so people don't necessarily have to go in to Glasgow or Edinburgh, and this tour of Uncle Varick is perfect for that."
While fully supportive of Emans's production of Uncle Varick, Byrne has taken a back seat, leaving Emans to get on with things while he prepares for his forthcoming exhibition of new paintings at Edinburgh Art Festival.
"I don't have a minute to spare," says Byrne. "I'm working on number 12 at the moment, and they need about 25. I know I have to come up with the goods, and not just any old goods. That's why I'm happy just to let Michael get on with it, and I'll be going with an open mind. I want to be surprised and delighted."
Such generosity and openness of spirit are the key to Byrne's writing.
As Emans observes, "John has such a hugely colourful, artistic style of writing that's so exciting and uniquely Byrne, the way he gives clues in the text to how a line should be said. It's so vibrant, but above all what stands out is just how much he gets the human condition."
Uncle Varick, Village Theatre, East Kilbride, Wednesday; Howden Park Centre, Livingston, Thursday; Eastwood Theatre, Giffnock, Sunday. The tour continues throughout May and June.