Deep in rural Herefordshire, where the SAS has its headquarters, is a place called the Killing House.

It’s a nondescript building used to train special forces soldiers for rescue operations, where the objective is to extract a human “hostage” alive while eliminating dummy terrorists. When David Davis, a former special forces reservist, visited his old regiment recently, he was invited to try the exercise for old times’ sake. The secretary of state for exiting the EU, now 68, inevitably couldn’t resist.

He emerged 10 minutes later, so the story goes, grinning. Targets dead, hostage unharmed, mission accomplished.

It’s a classic Davis story: deeply macho, somewhat swanky (he was once described as a man who can even swagger sitting down), almost too good to be true. But it illustrates his inability to decline a challenge, now being sorely tested.

His friends swear he’s focused solely on his ministerial job, but it is no longer ridiculous to suggest David Davis could be prime minister by the end of summer. The cabinet remains desperate to avoid another leadership contest, but Tory MPs are questioning whether a clearly miserable Theresa May can get through Brexit. (She is said to have told friends this is the hardest period of her life since her parents’ death.) Davis is the grassroots favourite to succeed her, according to the website Conservative Home’s recent poll of activists, while becoming increasingly central to propping her up.

“She depends on him; it’s almost embarrassing to watch,” says a source who has observed their meetings. “All the decisions are now made by Theresa May, Philip Hammond and David Davis. The civil service has worked out that if the three of them get in a room and agree they can bring the cabinet with them. But she defers to Davis – she’s rude to Philip Hammond.” Were she to quit halfway through Brexit negotiations, Davis could offer a seamless transition, if backed by his increasingly good friend the chancellor.

He has grown in office, too. Once breezily overconfident about Brexit, he has collided with reality, as presented by his civil servants, Hammond, business contacts and his Remain-supporting special advisers, James Chapman and Raoul Ruparel, and proved surprisingly willing to listen.

His pitch would be as the Leaver both sides can work with; willing to compromise where it saves jobs, liberal on immigration, a natural deal-maker whose hands were previously tied by an inflexible Number 10 operation. (Chapman, who recently left government, has publicly suggested that May’s crowd-pleasing insistence on leaving the European court of justice is seriously obstructing progress.) Some now think he could be persuaded into a “soft-landing” Brexit, cushioned by a transition period of dealing with the EU on similar terms to those prevailing now.

And if Davis is unlikely to serve far past the age of 70, that suits junior ministers impatient for their turn perfectly well. “Young cardinals,” observes an ally, “like old popes.”

But it remains unclear whether he has overcome his lifelong weakness – a tendency to overplay a good hand.

Davis was born in York in 1948, the son of a single mother in an age when that was a source of shame. His father, a married man, effectively abandoned his mother on discovering her pregnancy and the young David was initially raised by his grandparents before joining his mother and her new husband in their London council flat.

A decade ago, these humble origins established Davis’s credentials as a Tory moderniser, despite some haziness about how exactly he thought the party should change. But now, the tale of how he hauled himself up by his bootstraps, via grammar school, a redbrick university, then the London Business School and Harvard and, eventually, a successful business career with the sugar company Tate & Lyle, resonates for other reasons. The next Tory leader must find answers to Jeremy Corbyn’s rage against the establishment. Knowing what it’s like to grow up without money, to feel like an outsider, matters.

He has a ready temper in which some discern chippiness, although it’s perhaps fairer to say that he has much restless energy to discharge. (In opposition, he’d spend summers walking coast to coast across England, getting the frustration out.)

His passion for civil liberties is real, reflecting an equally fervent belief in a smaller state

But Davis is naturally gregarious and bouncy, an incorrigible gossip with friends across the political spectrum; Nick Clegg, Alex Salmond and Alastair Campbell have all helped him understand their respective parties. He can be excessively laddish – he was recently caught texting a friend that suggestions he jokingly tried to hug Labour’s Diane Abbott were wrong because “I am not blind” – but enjoys a long and devoted marriage to Doreen, whom he met at university and with whom he has three grownup children. Weekends at home in Yorkshire with her are sacrosanct and Davis spends a lot of time with one of their grandchildren, who is severely disabled.

He is, however, no soft touch. First elected to parliament in 1987, he earned his spurs in John Major’s whips’ office during Maastricht (where his strong-arm tactics earned him the nickname “DD of the SS”) and was promoted to minister for Europe.

But after Major lost the election, he spurned the opposition frontbench to chair the Commons public accounts committee, a usefully wide insight into departments across Whitehall – and nurse wider ambitions.

When he ran in the 2001 leadership contest, he trailed Iain Duncan Smith, Michael Portillo and Ken Clarke by some distance. But for a while, it seemed 2005 might be his moment. He was “sound” enough on Europe for rightwingers, but thoughtful enough not to frighten centrists, while elderly Tory members loved his story of self-made success. His team set out to dominate the race, railroading MPs into believing he’d win so they might as well get on board. But the aggression of some of his lieutenants caused resentment and, privately, too many supporters seemed more resigned to his victory than eager for it. It was as if they looking over his shoulder for something better and then, in October, found it. All five contenders were due to make speeches, never Davis’s forte, at that autumn’s party conference. David Cameron’s is remembered as the point his party sat up and noticed him but Davis never recovered from a dull, halting effort – and there, it was assumed, his career had peaked.

But he had one last surprise up his sleeve. Accepting a job as Cameron’s shadow home secretary, he forged unlikely but effective alliances with Liberty’s then director, Shami Chakrabarti, and the Labour left to attack what he saw as Tony Blair’s deeply illiberal counter-terrorist measures. Among those he got to know was one Jeremy Corbyn, who, in 2015, accompanied him and fellow Tory MP Andrew Mitchell to Washington to urge the release of a Briton imprisoned in Guantánamo Bay. His passion for civil liberties is real, reflecting an equally fervent belief in a smaller state. “He’s just a proper Tory,” says a friend. “He’s sceptical about giving the state powers which it doesn’t need.”

But the plaudits he won for those principles may well have gone to his head. In 2008, amid controversy over Gordon Brown’s plans to detain terror suspects for up to 42 days without trial, Davis stunned his party by effectively calling a byelection against himself. He quit parliament, urging constituents in Haltemprice and Howden to re-elect him if they agreed with him on civil liberties, in what he portrayed as a blow for freedom.

Most colleagues considered it pointless at best, mad at worst, and Cameron did not reappoint him to the frontbench when he was re-elected. Even Davis was surprised when last year Theresa May brought him back from the dead. Her inner circle still regard that as one of their better decisions, despite Davis’s subsequent advice to go for what would become a disastrous snap election. They were desperate for Leavers heavyweight enough to fill key Brexit posts and Davis had learned much from the Major years both about Europe and handling a tiny parliamentary majority; he thinks strategically and has sharp political antennae.

Which may be why it’s Davis, not Liam Fox or Boris Johnson, who seems to have caught a shifting mood among Brexiters; a fear that their grand project is in trouble, although they blame government’s failure to do it justice, not Brexit itself.

The worry, says a source, is that negotiations will be bloody and government will have to roll with some serious punches. “We’re going to have to hold our nerve, ride out public opinion turning against it.’’ It’s a role tailormade for someone, like Davis, impervious to self-doubt.

But even if Theresa May stays the course, his influence over Brexit – that pivotal moment in national life – is clearly growing and with it a daunting responsibility. “This is his last chance,” says a colleague, “and he knows he can’t fuck it up.”

Mission not accomplished yet.

THE DAVIS FILE

Born David Michael Davis in York, on 23 December 1948; he was brought up in south London. Studied molecular science/ computer science at Warwick and later studied business at London Business School. Married to Doreen; three children.

Best of times He made his name as an effective chair of the public accounts committee when William Hague was leader of the Conservative party. Now he is emerging as one of the strongest voices in government, as Brexit secretary, recognised last week as he topped a popularity poll of party activists conducted by website Conservative Home.

Worst of times Though he was an early favourite to win, he performed badly during the 2005 Conservative leadership contest, losing out to David Cameron.

What he says

“So how will this look if we get it right? We will have a more dynamic economy, trading throughout the world. Our businesses will have greater global opportunities and will be more competitive.”

What they say

“He’s just a proper Tory. He’s sceptical about giving the state powers it doesn’t need.” One of Davis’s friends