TORY MP Nigel Evans has called for Theresa May to step down as Prime Minister "today".
The joint executive secretary of the 1922 Committee told the BBC Today programme the Conservatives need new leadership and that the process "can't start soon enough".
Evans said: "To be honest I would be delighted if she announced today she was announcing her resignation and we could then have an orderly election to choose a new leader of the Conservative Party.
"I believe 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."
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The Ribble Valley MP said he had reflected on the situation over the Easter break and said the "severe problems" over Brexit meant May had to go "as soon as possible".
He said: "I hope she does accept the fact the call for her resignation now is growing into a clamour.
"It's not now just within Westminster – we've heard the news that 70 chairs of Conservative Associations throughout the country have now said that they reluctantly have no confidence in the Prime Minister."
Evans said, with 40% of Conservative councillors likely to support Nigel Farage's Brexit Party in the European elections, May's position would be "totally untenable".
When asked when May should step down, he added: "Preferably today.
"We need fresh leadership of the Conservative Party as quickly as possible – it can't start soon enough."
Prisons Minister Rory Stewart has backed Theresa May as "our best hope" of resolving Brexit.
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He told the BBC Today programme "the problem is not the Prime Minister, the problem is Brexit".
He said: "I think she is doing a very, very important job and she needs to get the first stage of Brexit done.
"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.
"It's nothing to do with the individual, it is that people disagree deeply over Brexit."
Stewart acknowledged "a lot of MPs are cross and impatient" but insisted May should keep her job.
He said: "Going into a leadership battle with Brexit completely unresolved it would not only have enormous uncertainty for the country, but it would tear politics apart, because it would lead to this deep, deep polarisation between people who want to remain and people who want to leave with no deal and she is our best hope of resolving that."
The Penrith MP added: "People are somehow suggesting that she is lying, that she never wanted to do this.
"She has been absolutely committed to doing these things and I think the public and everybody has seen painfully why it hasn't happened, not through any will of her own but simply because we don't have a majority in Parliament.
"Attacking a prime minister to say 'you wanted to do this'... if she simply can't get the votes to do it, is not a criticism of her but a criticism of the people in the Conservative Party who won't vote with her."
Brexit Party leader Nigel Farage has pledged to target Labour voters.
Speaking at a campaign event to announce five new candidates, he said the party was already "doing very well" with Conservative supporters.
The former Ukip leader said: "There are five million people that voted for Jeremy Corbyn and voted for Brexit as well and that's going to be our task.
"I think we will go on squeezing the Conservatives and squeezing Ukip down to virtually nothing.
"We're going to go after that Labour vote in a very big way."
Boris Johnson's sister Rachel is to stand as a European election candidate for Change UK, the pro-Remain party set up by The Independent Group of MPs.
Johnson, who will stand in the South West region, announced her move at an event to launch the party's election campaign in Bristol.
She was joined on stage by former BBC Newsnight presenter Gavin Esler, who is also standing.
Esler, who is standing for the party in London, slammed Nigel Farage and Jacob Rees-Mogg as "pretend men of the people".
He described them as the "posers of Brexit" and said they were "selling the same old snake oil".
"They claim to speak for the people – they do not," Esler said.
"They stole our patriotism and I want it back."
Esler, who has more than 40 years' experience in journalism, said: "I have never been a member of a political party but I am now.
"I have never been a candidate in an election but I am now.
"I have never been seriously worried about the future of our country but I am now.
"Our political system is a joke. It is a worldwide joke.
"They are laughing at us – not with us, at us."
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