Theresa May is holding talks with senior Conservative MPs who are demanding she sets a date for her departure from Downing Street.
The prime minister, under growing pressure from her own MPs to quit, is meeting the executive of the backbench 1922 Committee to discuss her future.
Existing rules mean she cannot be formally challenged until December.
Mrs May has said her departure depends on delivering Brexit, but her plan has been rejected by MPs three times.
She will attempt to push her plans through the House of Commons again in the week beginning 3 June, when MPs will hold another vote on the withdrawal bill – needed to implement the Brexit deal the prime minister reached with the EU.
Asked whether she would resign if the bill was rejected a fourth time, Mrs May said it would “ensure that we deliver Brexit for the public”.
During the talks with the prime minister, the 1922 Committee’s executive is expected to press her to set out her departure timetable, regardless of whether Parliament backs her Brexit withdrawal agreement.
Afterwards, the executive will hold another meeting where changes to the leadership rules could be discussed again.
Last month, the 1922 Committee executive narrowly decided against changing the party’s leadership rules to allow an early challenge to Mrs May.
Theresa May’s very long goodbye
Twice my colleague in Paris asked the prime minister if she would resign if her Brexit plan is rejected by MPs again. Twice she completely swerved the question.
You can’t blame her for wanting to avoid it. Theresa May is not exactly fond of revealing much in interviews in any case.
And for her to go anywhere near tangling with her potential exit would have breathed even more fire into the frenzied speculation.
But it is notable that the prime minister didn’t even repeat any of her rehearsed lines about her promise to see this phase of Brexit through, and then go.
Pressure has grown on Mrs May since the Conservatives’ local election drubbing two weeks ago. Much of the anger in the parliamentary party is focusing on Mrs May’s talks with Labour, aimed at reaching a cross-party compromise to get her deal through the Commons.
Local Tory associations have confirmed they will hold a vote of confidence in her leadership on 15 June, although its result will not be binding.
Eurosceptic MP Peter Bone produced a letter from Conservative activists in the Commons on Wednesday, saying they had lost confidence in the prime minister and want her to resign before European elections on 23 May.
They view Mrs May deal with the EU as “worse than staying in the European Union”, he said.
Mrs May’s former chief of staff, Nick Timothy, wrote in his Daily Telegraph column that the prime minister must “accept that the game is up”, and quit to avoid a “national humiliation” and save the Conservative Party.
Sir Geoffrey Clifton-Brown, treasurer of the 1922 Committee, said: “What I would like to see is her set out a timetable to trigger a leadership contest.”
He said it would be “infinitely preferable if she set a date rather than us force her out”.
Prominent Brexiteer Mark Francois said a predicted poor performance by the Conservatives in next week’s European Parliament elections would increase the pressure on Mrs May to stand down.
When the results are announced, “my colleagues will finally wake up and smell the coffee if they have not, indeed, done so already,” he added.
But International Development Secretary Liam Fox told BBC Radio 4’s Today programme the prime minister’s pledge to stand down when the Commons supports her Brexit withdrawal deal showed “integrity” as she was “willing to make that personal sacrifice to get the policy through”.
He said the public had instructed MPs to deliver Brexit in the 2016 referendum, asking: “Why have they not carried it out?”
Brexit Secretary Stephen Barclay told MPs their concerns had been listened to and that was “reflected in the draft legislation that is being prepared”.
But shadow Brexit secretary Keir Starmer said Labour opposed the idea of passing the bill without an agreed cross-party deal, as it would “put the cart before the horse”.
He questioned whether the bill was “about keeping the prime minister in office for another week, to give her a lifeline” ahead of the 1922 Committee executive meeting.