Talks between the government and Labour on Brexit will resume later as MPs return to Westminster following the Easter break.
Cabinet ministers will meet senior opposition figures in an attempt to solve the impasse by finding a deal that could win the support of MPs.
But some Tory MPs are angry the talks with Labour are even taking place.
Leading backbencher Nigel Evans called on Theresa May to step down as prime minister “as soon as possible”.
The joint executive secretary of the back bench 1922 Committee and MP for Ribble Valley told BBC Radio 4’s Today programme: “The only way we’re going to break this impasse properly is if we have fresh leadership of the Conservative Party…
“If there was an announcement today by the prime minister then of course we could start the process straight away.”
He said Mrs May “had been reaching out to the Labour Party and Jeremy Corbyn, when she should have been reaching out to the people”.
But Prisons Minister Rory Stewart said Theresa May was doing a “good job” and deserved “praise not blame”.
“The idea somehow that some new fresh leader with extraordinary charm and nimble feet would be able to suddenly get the deal across the line is mistaken,” he told the BBC.
“It’s nothing to do with the individual, it’s that people disagree deeply about Brexit.”
Mr Evans’s comments came after it emerged that Mrs May faces a no-confidence challenge from Tory campaigners.
More than 70 local association chiefs have called for an extraordinary general meeting to discuss her leadership and a non-binding vote is to be held at the National Conservative Convention EGM in May.
If the grass-roots Tory vote showed a lack of confidence – it could put greater pressure on the 1922 Committee to find some way of forcibly removing the PM from office.
That pressure could increase further if the Tories poll badly in local elections on 2 May.
Under current party rules, MPs cannot call another no-confidence vote until December 2019 – but when senior members of the 1922 meet on Tuesday afternoon, they will discuss whether steps should be taken to try to change them.
Mrs May is due to chair a cabinet meeting on Tuesday morning, and her de facto deputy, David Lidington, will attend the talks with Labour later.
Cross-party meetings have been going on for a number of weeks after Mrs May’s EU withdrawal deal was rejected for a third time by MPs.
But BBC assistant political editor Norman Smith said most people at Westminster seem to take the view that the prime minister is clutching at straws by pinning her hopes on those talks.
He said Labour’s position has not changed and Mrs May will have take a political decision to accept a customs union with the EU if she is to get Jeremy Corbyn to sign off her plans.
If she does that, our correspondent added, it would almost certainly provoke a cabinet walkout and open warfare on the backbenches.
In separate news, Change UK will launch its European election campaign in Bristol, while Nigel Farage’s Brexit Party will unveil more of its candidates in London.
Change UK is made up of 11 former Labour and Tory MPs who quit their parties in February.
The UK has been given an extension to the Brexit process until 31 October.
This means the UK is likely to hold European Parliament elections on 23 May.