During three-hour crisis meetings at Chequers on Sunday, Mrs May was apparently warned by Brexiteers Iain Duncan Smith and Jacob Rees-Mogg that she must set out a timetable for her departure to get her embattled deal through the Commons this week.
But the Prime Minister is said to have dug in, warning Brexiteers who attended the meetings including David Davis and former foreign secretary Boris Johnson that if they refused to get behind her plan, MPs would try to force through a “soft” Brexit.
Labour's deputy leader Tom Watson said there were rumours that "the Brexiteers at Chequers have secured the PM's commitment to resign in return for their support for her deal".
Writing in the Telegraph, Mr Johnson said the Government has “bottled it completely” over Brexit by asking to delay and urged Mrs May to “channel the spirit of Moses” and “tell Brussels” to “let my people go”.
At the start of another crunch week in Westminster, the Commons is due to vote on an amendment which would force a series of indicative votes on alternatives to the Prime Minister's Withdrawal Agreement.
Defeat for the Government on Monday night on the plan, tabled by former ministers Sir Oliver Letwin and Dominic Grieve and Labour MP Hilary Benn, would be a further humiliation for Mrs May.
The proposal seeks to pave the way for a series of indicative votes in the Commons on Wednesday, effectively taking control of the Brexit process out of the hands of the Government.
Brexit Secretary Steve Barclay warned the risk of a general election would increase if MPs took control of parliamentary proceedings and brought about a "constitutional collision".
But Chancellor Philip Hammond said "one way or another" MPs would be given the opportunity this week to decide what it is in favour of, though could not confirm whether Tories would be given a free vote on the options.
After a weekend which saw two senior ministers Michael Gove and David Lidington dismiss reports of a "coup" to oust the PM, Mrs May will convene her Cabinet in the morning before she updates the Commons on the Brexit process following last week's European Council summit where she agreed to delay Britain's departure beyond March 29.
On Sunday, Mrs May held "lengthy" talks with prominent Brexiteer backbenchers including Mr Johnson, Mr Rees-Mogg and Mr Davis at her country residence Chequers.
A Downing Street spokesman said: "The PM and a number of Government ministers met today at Chequers for lengthy talks with senior colleagues about delivering Brexit.
"The meeting discussed a range of issues, including whether there is sufficient support in the Commons to bring back a Meaningful Vote this week."
Mr Johnson claimed the Government had "chickened out" of delivering Brexit this week and told Mrs May to set out "convincing proofs" of how the next phase of the negotiations will be different from the last to win support for her deal.
In his column for the Telegraph, he said: "If she cannot give that evidence of change - she should drop the deal, and go back to Brussels, and simply set out the terms that so many on both sides - remainers and leavers - now believe are sensible.
"Extend the implementation period to the end of 2021 if necessary; use it to negotiate a free trade deal; pay the fee; but come out of the EU now - without the backstop. It is time for the PM to channel the spirit of Moses in Exodus, and say to Pharaoh in Brussels - let my people go."
However, Foreign Office Minister Mark Field said he would support revoking Article 50 if it became an option in the event Mrs May's deal was defeated and free votes granted for indicative votes.
He told BBC Radio 4's Westminster Hour: "My personal view is that I would be happy to revoke Article 50 - I appreciate that is probably a minority view - but if we get to this utter paralysis and I sincerely hope that in the next 48, 72 hours we do not, then if that becomes an option that's an option that I would personally take."
Meanwhile, The Sun newspaper used its front page on Monday to urge Mrs May to promise to resign in order to win support for her deal from Tory Brexiteers and the DUP.
"Unlike so many she has determinedly respected the will of the Leave majority... She must now take the next principled step - and show she is not just another craven politician determined to cling to power," it said.