Theresa May has insisted it is still possible for the UK to leave the European Union on 29 March if enough MPs back a revised withdrawal deal, amid signs hardline Eurosceptics may be softening their demands.

In an article in the Daily Mail, the prime minister pleaded with MPs to get behind her deal, after she was forced to give them votes on extending article 50 and ruling out no deal if her withdrawal agreement does not pass.

“Our absolute focus should be on working to get a deal and leaving on March 29,” she wrote. “Doing so would give businesses and citizens the certainty they deserve.

“By committing Labour to holding a second referendum, despite promising to implement Brexit, Jeremy Corbyn has shown once again that he cannot be trusted to keep his promises. His cynical political games would take us back to square one.

“Instead, parliament should do its duty so that our country can move forward.”

May is hoping to win round more hardline Eurosceptics to back her deal, playing on their fears that a delay to Brexit could mean a second referendum.

However, she needs a revised agreement from Brussels in the coming days on changes to the Irish backstop to prevent the UK being locked into a permanent customs union.

The European Research Group of MPs, pushing for a hard Brexit, had been clear that the backstop must be removed.

But their leader, Jacob Rees-Mogg, appeared to temper this demand in an interview on BBC Radio 4’s Today programme on Wednesday.

He said: “I can live with the de facto removal of the backstop, even if it isn’t de jure. What do I mean by that? I mean that if there is a clear date that says the backstop ends, and that that is in the text of the treaty, or equivalent to the text of the treaty - if it were to be an appendix to the treaty; bear in mind, the Irish backstop is in itself an appendix to the treaty. So if you had a further appendix that said, ‘This will not go beyond a particular date’, and a short date, not a long date, then that would remove the backstop in the lifetime of parliament. That would have a reasonable effect from my point of view ...

“A changed deal is a changed deal. Of course, I would be open to considering that.”

After a meeting of the ERG on Tuesday, Rees-Mogg also acknowledged 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.”

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.

Some Eurosceptic Tory MPs are particularly worried about an extension as they fear this could end with May reaching out to Labour to work on a softer Brexit deal they could vote for, a second referendum or general election resulting in a Labour victory.

In a stormy cabinet meeting on Tuesday at which May announced her plan, the chancellor, Philip Hammond, and the work and pensions secretary, Amber Rudd, both said any extension of the Brexit talks must be usedto find a new coalition in parliament.

The comments strongly suggest an extension is most likely to result in the softening of May’s proposed deal in order to bring Labour MPs onboard – an outcome hardline Tory Eurosceptics would want to avoid.

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 ministers had undermined the prime minister. The 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 said he believed their concerns had been appeased by May’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” to hold the series of votes.

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