Cabinet ministers have told Theresa May she must use any delay to Brexit to face down the Eurosceptic wing of the party and forge a consensus in parliament, as the prime minister finally conceded to offer a vote on extending article 50.

May reluctantly promised MPs they would have the chance to reject a no-deal Brexit next month and also offered a vote to extend article 50 in an attempt to stave off the threat of dozens of resignations from her frontbench.

In an earlier stormy cabinet meeting, where May announced her plan, the Guardian understands the chancellor, Philip Hammond, and the work and pensions secretary, Amber Rudd, both said any extension of the Brexit talks must be used “to find a new coalition in parliament”.

The comments, which are likely to further irritate Tory Eurosceptics, strongly suggest an extension is most likely to result in the softening of May’s proposed deal in order to bring Labour MPs on board.

In her statement to the Commons on Tuesday, May said she planned to hold the next meaningful vote on her Brexit deal by 12 March. If it was defeated again, it would be followed by a vote on 13 March on leaving with no deal and, if this was rejected, a vote on 14 March for an extension to article 50.

At the cabinet meeting, Andrea Leadsom, the Commons leader, was among those most angered by the decision to offer a vote on an extension and suggested some cabinet ministers had undermined the prime minister. Treasury secretary Liz Truss called the trio “kamikaze” ministers, according to several sources.

Rudd, along with the justice secretary, David Gauke, the business secretary, Greg Clark, and the Scotland secretary, David Mundell, warned the prime minister in private last week that she must rule out a no-deal Brexit or risk dozens of ministerial resignations.

Clark, Rudd and Gauke then wrote in the Daily Mail that Brexit should be delayed if a deal was not passed by mid-March. While Hammond did not sign up to the public statements from his ministerial colleagues threatening to defy Tory whips and back the Cooper-Letwin amendment, allies insisted the chancellor “does his lobbying inside the tent”.

Even the party’s chair and May loyalist, Brandon Lewis, said ministers should be cautious about the framing of no deal, pointing out the option was supported by many Tory members. Rudd told Lewis, however, that the cabinet “needed to be straight with the British people”.

“This is a bitter pill to swallow,” one cabinet source said. “But ultimately we felt like we had been left with no other option. There has been a private back-and-forth. Should we just let people resign? But that causes its own party problems, too.”

The promise to vote on a Brexit delay is likely to avert a serious ministerial rebellion. Dozens had been privately threatening to back an amendment on Wednesday to the government’s latest Brexit motion, which would have paved the way for legislation to extend article 50 beyond the 29 March deadline.

The amendment had been tabled by the Labour MP Yvette Cooper and the Tories Nick Boles and Sir Oliver Letwin, but the latter pair said they believed their concerns had now been appeased by the prime minister’s pledge to hold the three votes in mid-March.

The cross-party coalition led by Cooper still tabled three amendments to the government’s motion, but was likely to push only one of them to a vote. It sets out the promises made by May on Tuesday in a form that MPs can vote for, giving them added reassurance that neither the prime minister nor other cabinet ministers will go back on those promises.

Cooper said the amendment would “pin down and confirm the commitment made by the prime minister today” to hold the series of votes.

The coalition was awaiting further reassurances from ministers on Wednesday, including from the Brexit secretary, Steve Barclay. If it sensed any backsliding, it could still push for a vote on the two other amendments.

May’s spokesman said no decisions had been made on whether the government would support an extension and the prime minister would focus instead on securing a deal MPs could support.

Cabinet sources suggested any decision on whipping MPs to vote a certain way would only be made once the meaningful vote had been lost. “Why would we take a decision on whipping before it is absolutely necessary?” one cabinet source said. “You are only going to inflame people on either side.”

Despite the apparent win for the soft Brexiters in cabinet, government officials remain anxious that a no-deal Brexit may simply be delayed by a couple of months, with a second extension highly unlikely. “You can probably only pull that cord once,” a Whitehall source said.

After a meeting of the Brexit hardliner European Research Group, MPs said they still held out hope that the attorney general, Geoffrey Cox, could deliver a suitable compromise on the Irish border backstop to allow MPs to back May’s deal.

The group’s chair, Jacob Rees-Mogg, said the vote “doesn’t change anything enormously … it remains the position that we hope that the prime minister will be able to secure changes we will be able to accept”.

He said the group recognised that there was unlikely to be the parliamentary numbers to leave with no deal. “There is no enthusiasm for backing the deal as it is,” he said. “If the Irish backstop could be removed or fundamentally altered, then that would be a different kettle of fish.”

Cox arrived in Brussels on Tuesday for talks with the EU’s chief negotiator, Michel Barnier, where he is expected to pressfor legal guarantees that the backstop remains a last resort and temporary option.