Cabinet minister Andrea Leadsom has said she is “seriously considering” standing for the Conservative leadership.
Ms Leadsom’s comment came as pressure increased on Theresa May to name a date for her departure and cross-party Brexit talks with Labour dragged on without a conclusion.
Talks on a compromise deal were continuing in Whitehall, with Labour insisting the Government still needs to shift on its red lines.
Discussions on Tuesday were described as “constructive and detailed” by Downing Street and “very robust” by Labour, with shadow business secretary Rebecca Long-Bailey saying: “Nothing has been agreed yet.”
Senior Tory backbenchers on the executive of the 1922 Committee were due to meet in Westminster to discuss the outcome of a meeting at which chair Sir Graham Brady is believed to have set out to Mrs May MPs’ concerns about her position in the wake of last week’s disastrous local elections.
The 1922 executive held back last week from approving a change in rules to allow Mrs May to face a no confidence vote from MPs before December.
But Sir Graham made clear then that the group wants “clarity” on Mrs May’s plans to step down.
The Prime Minister has promised to quit once her Withdrawal Agreement is ratified, but has not made clear what she will do if it fails to pass through Parliament.
And with her effective deputy David Lidington suggesting he hoped the deal can be concluded by July, there is speculation she may seek to hang on until the annual party conference in the autumn.
Ms Leadsom, who stood for the leadership in 2016 but pulled out to give Mrs May a clear run on the job, became the latest senior Tory to indicate she may throw her hat into the ring for the upcoming contest.
The Leader of the Commons told ITV1’s Good Morning Britain: “I’ve supported her for the last three years to get Brexit over the line. She has said she’s going, so yes I am seriously considering standing.”
Ms Leadsom warned that the rise of the Brexit Party threatened to put Jeremy Corbyn’s Labour in power by costing the Tories seats in a general election.
“That’s something that keeps me awake at night, the prospect of a Marxist government coming in by the back door because people are so sick of the Brexit issue,” she said.
“That’s why we absolutely have to leave.”
Ms Leadsom described Mrs May’s Brexit agreement as “tolerable”, but said she would be prepared to leave the EU without a deal.
“I don’t say that no-deal is better than the Prime Minister’s deal.
“I think the Prime Minister’s deal is the best solution because it protects jobs and supply chains,” said the Commons leader.
“What I’m saying is at the same time I don’t think no-deal would be the disaster some people portray it as.”
Meanwhile, Brexit Party leader Nigel Farage told GMB that Mrs May should go: “Tory MPs have not got the backbone to get rid of her. They could have got rid of her last December, they should have done.
“She is without doubt not just the worst Prime Minister in my lifetime but the most dishonest as well.”
International Trade Secretary Liam Fox strongly rejected the idea of the UK remaining in the EU’s customs union, one of Labour’s key demands in the Brexit talks.
At a trade conference in London, he said it would be “bad for Britain” and leave access to the UK’s markets as a “commodity” to be traded by Brussels.
“The EU would be able to make access to the UK market part of their offer in any trade agreement and we would find ourselves in a unique position in our trading history in that we would be being traded.
“We would be a commodity in that particular agreement, where the EU would be able to offer access to the UK as part of their offer,” he said.
“It’s a situation that would leave the UK as a rule taker and in terms of our ability to shape trade policy would probably leave us in a worse situation than we are today, inside the EU.”
Senior Tory activists will consider the question of Mrs May’s leadership at an emergency meeting of association chairmen, set for Saturday June 15.
The treasurer of the 1922 Committee, Sir Geoffrey Clifton-Brown, told the Press Association he would back their bid to force Mrs May out with a no-confidence vote next month if she did not go of her own volition.
The Tory MP for the Cotswolds said it was the Prime Minister’s decision on when she should go but he would “absolutely” support grass roots moves to try to topple her in June if her departure date was not imminent.
When asked what should happen if Mrs May failed to set out her own timetable for departure, he said: “It begins to get much more messy.
“It would be much easier and I think the European elections would be much easier if she did set out her own timetable to go but it is up to her.
“I think it’s quite possible they (grass roots members) might vote for no-confidence in June.”