Cabinet ministers will make it clear they believe Theresa May should step down after the local elections in May and allow a new leader to deliver the next phase of the Brexit negotiations, the Guardian understands.

Senior figures in government have suggested they want the prime minister to leave shortly after the first phase of the Brexit negotiations finishes – or risk being defeated in a vote of no confidence at the end of the year.

May wants to stay in place for long enough after Brexit to secure a political legacy beyond the fraught negotiations. But some ministers believe she should announce the timeline for her departure “on a high” after the local election results, paving the way for a Conservative leadership contest over the summer.

Brexiters in the cabinet are keen to see a new leader take over for the next stage of the negotiations with the EU, which May has already pledged will involve more active involvement for politicians rather than advisers.

The hardening mood among cabinet ministers on the timeline for her departure will place further pressure on May before a critical week of Brexit talks and votes amid a febrile climate in Westminster.

On Thursday the Guardian revealed that remainer ministers emboldened by the departure of three MPs to the Independent Group (TIG) were threatening to rebel against her leadership to prevent a no-deal outcome – daring her to sack them.

And in a fresh blow to May, three cabinet ministers publicly say they would back moves to delay Brexit if she fails to get her deal through parliament.

In a joint newspaper article, Amber Rudd, the work and pensions secretary, David Gauke, the justice minister, and the business secretary, Greg Clark, say they want to ensure the UK does not crash out of the EU without a deal on 29 March. And they insist they are prepared to defy the prime minister and join those MPs pushing for an extension to article 50 if there is no significant progress next week.

Writing for the Daily Mail on Saturday, they argue that a no-deal Brexit would wreck the country’s economy and put its security at risk. “If there is no breakthrough in the coming week, the balance of opinion in parliament is clear – that it would be better to seek to extend article 50 and delay our date of departure rather than crash out of the European Union on 29 March,” they write.

“It is time that many of our Conservative parliamentary colleagues in the ERG recognised that parliament will stop a disastrous no-deal Brexit on 29 March. If that happens, they will have no one to blame but themselves for delaying Brexit.”

Ministers who want May to go are confident that if Brexit can be delivered on time in March, the party should be able to secure some promising results in the local elections that would provide a face-saving context for her early departure. Any extension to article 50 beyond 29 March would be likely to scupper the preferred timeline.

May pledged to Conservative MPs before the confidence vote in December that she would stand aside before 2022, though later made it clear she intended to stand should there be a snap election.

Cabinet ministers had hinted they would not like to see May take charge of the spending review later this year – which would set the direction of departmental spending until the next election.

The prime minister has a year’s grace before another confidence vote could be called. Should she refuse to go this year, at least one cabinet minister has said they believe she would be ousted by another confidence vote if one were called at the end of the year. Her position could become untenable sooner if enough senior colleagues were to resign or publicly express their dissatisfaction with her leadership.

Although Brexiters inside and outside of the cabinet, as well as former remainers who have now enthusiastically embraced leaving, believe new leadership is needed for the next phase of negotiations, it is unlikely that they will coalesce around a single candidate.

Jeremy Hunt, Sajid Javid, Liz Truss and Penny Mordaunt, as well as Boris Johnson, David Davis, and Dominic Raab are all likely to want to test their support bases among MPs, whereas Tories from the softer wing of the party are more likely to coalesce around one name, Amber Rudd. However, a Brexiter is likely to be far more popular with the membership.

If a leader with the backing of the hard Brexit-supporting European Research Group were to take charge, there are concerns that MPs on the left of the party could defect to TIG.

May is understood to have taken some convincing that she needed to pledge to step down in order to defeat the confidence vote held before Christmas and is keen to undertake a “domestic reset”.

However, the legislation involved for parliament’s second session view makes it difficult for the prime minister to make significant changes, with much of the domestic agenda already announced, including May’s flagship legislation on domestic violence.

The prime minister faces potential defeat in parliament next Wednesday, with scores of backbenchers and up to 25 ministers prepared to vote for an amendment tabled by Yvette Cooper and aimed at securing an extension to article 50. May has promised to table a motion in the House of Commons on Wednesday – which MPs will be able to amend – if she has not been able to put a revised withdrawal agreement to parliament by then.

She will fly to an EU-Middle East summit on Sunday, where she will hold bilateral meetings with senior figures including the European council president, Donald Tusk, as part of attempts to secure “legally binding” changes to the backstop.

Downing Street said the prime minister had spoken to 26 of the EU’s 27 leaders in the past fortnight, as she tries to convince them to make changes she can sell to MPs.

“She will have a period of engagement again, on Sunday, Monday, with EU leaders,” a spokesman for the prime minister said. “She has said it is not easy”.

The Brexit secretary, Steve Barclay, and the attorney general, Geoffrey Cox – whose legal advice on the backstop is critical to convincing Brexiters to support the deal – will return to Brussels on Monday, as technical talks between the two sides continue.

Privately, though, senior government sources played down the likelihood of any deal being reached in the next few days that May could put before MPs.

A spokeswoman for May said: “We are working very hard to bring it back as soon as possible … Work is continuing at pace to achieve the changes we need.”