Theresa May is facing pressure from rebellious Tory Brexiteers to stand down by the end of June after a crunch meeting.
The Tory 1922 committee was understood to be split down the middle over changing the rules to make it easier to oust the Prime Minister.
Ahead of the meeting backbench chief Sir Graham Brady met Mrs May to tell her many MPs want her to set a departure date.
In a high stakes move, the PM is considering flagship Brexit legislation which would give MPs a decisive vote as early as next week.
Downing Street said it was ready to introduce the withdrawal agreement bill even though MPs have rejected the PM’s plan three times.
But No 10 suggested they would only do so if there was some chance of Labour allowing it “secure passage” through the Commons.
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One source said: “If we know they are going to shut it down immediately then we’d think twice - but then also we’re running out of time.”
They acknowledged the bill would almost certainly be amended by MPs to give MPs a vote on a customs union and a second referendum.
It could give MPs one last chance to get Brexit through and avoid holding European elections on May 23.
But the gamble highlights the desperation of Mrs May’s top team to put her on the front foot over Brexit amid fierce Tory criticism.
Brexiteer members of the 1922 executive have called for a rule which prevents her from facing another confidence vote from her own MPs until December to be scrapped.
They are furious at Mrs May for delaying Brexit to October 31 and entering talks with Labour that could lead to a ‘softer’ deal.
A full meeting of the 1922 committee of Tory backbenchers will take place on Wednesday where MPs are expected to back calls for a rule change.
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Senior Tory Nigel Evans said: “It is with great reluctance that I’ve now come to the conclusion myself that we need somebody new at No 10 to take us to the next stage”.
But Remainer MPs on the executive, who want to keep the PM in place, criticised colleagues for speaking out over the potential change before it had been agreed.
Tory sources said one proposal under discussion would involve all 313 Tory MPs being balloted on the rule-change.
Brexiteers believe that, if passed, the PM would then see the writing on the wall and stand down.
Grassroots Tories are also plotting to force the PM out.
Mrs May faces a moment of acute danger after local elections on May 2 and EU elections on May 23, where the Tories are expected to be badly hit.
Cross-party Brexit talks with Labour resumed yesterday but neither side appeared hopeful they would be conclusive.
Government insiders believe they could break down later this week.
Each sides blamed the other for the continued deadlock with the PM suggesting to her Cabinet that Labour was dragging its feet.
Mr Corbyn accused the PM of “regurgitating” failed plans that had been rejected three times by Parliament.
