Michael Gove has been offered the job of Brexit secretary in the wake of Dominic Raab’s resignation, as Theresa May battles to shore up her authority, but he is demanding a shift in the government’s negotiating strategy first, Whitehall sources say.

Quick Guide How would the Tories trigger a leadership contest? Show No-confidence proceedings Forty-eight Conservative MPs would need to back a no-confidence vote in Theresa May to trigger a leadership contest, according to party rules. There are two ways a contest can be triggered, most obviously if the leader of the party resigns. If they do not, 15% of Conservative MPs must write to the chairman of the 1922 Committee of backbench Tories. With the party’s current crop of 317 MPs, 48 would be needed. After David Cameron announced his resignation, five Tory MPs stood for the leadership. Unlike Labour party rules, under which candidates go to a ballot of members as long as they have the support of 15% of the party’s MPs, Conservative candidates are whittled down to a final two before party members have their say. The ballot is based on “one member, one vote”, but in 2016 one of the final two candidates, Andrea Leadsom, withdrew from the race after a damaging interview with the Times about the fact that May did not have children. Her withdrawal meant May was made party leader without having been elected by members.

As the most senior figure from the Vote Leave campaign still inside May’s deeply-divided cabinet, Gove’s backing is regarded as pivotal to her efforts to keep her Brexit deal alive.

But the environment secretary is said to be urging the prime minister to shift her stance, and allow negotiators to go back to Brussels in a last-ditch bid to secure a deal that could be backed by parliament.

Gove’s close ally, Conservative backbencher Nick Boles, has called for the UK to try to secure a close, Norway-style relationship in the short term – and then negotiate a free trade deal.

Former Gove adviser Henry Newman claimed on Thursday that Raab had been frozen out of the Brexit process in recent weeks, with drafting changes sprung on him at the last minute.

6) Raab became a Brexit secretary in office but not in power. No wonder he personally felt he had to resign. May is entitled to give a free hand to her officials and Sherpas but don't expect Brexit ministers to stay on board in those circumstances — Henry Newman (@HenryNewman) November 15, 2018

Gove was reported to have spoken up in support of May’s deal at Wednesday’s gruelling cabinet meeting, where most pro-Brexit ministers express scepticism.

One senior Brexiter said it was highly unlikely May would accede to Gove’s demands, because she is now so closely associated with the Brexit deal.

The wrangle comes as Downing Street reels from a blizzard of resignations – including that of Raab – after Wednesday’s five hour-long cabinet meeting, at which she initially appeared to have secured her ministers’ backing.

Esther McVey resigned as work and pensions secretary on Thursday morning, as May prepared for a a three-hour marathon session in the House of Commons taking questions from MPs, many of them hostile interventions from her own side.

A growing number of backbench Conservative MPs, including the chair of the European Research Group, Jacob Rees-Mogg, are demanding a vote of no confidence in the prime minister’s leadership.

One cabinet source said: “We think she can survive two resignations. But if it gets to three she’s in trouble.”

Gove stood as Conservative leader after the referendum campaign in 2016, knocking his rival Boris Johnson out of the race, but losing out to Theresa May, who profited from the unseemly squabbling of her rivals.

Even if May is not challenged for the leadership in the coming days, it looks increasingly likely that she will fail to get her deal through parliament when she presents it to MPs, expected to be in early December.