Many of May's pro-Brexit Conservative members are angry after she said she would consider extending a transition period that kicks in after Britain exits the European Union on March 29.
Mrs May is facing demands to appear before the 1922 committee of Tory backbenchers on Wednesday to explain directly her negotiating strategy, but party chairman Brandon Lewis is now pencilled in to speak.
Theresa May will spell out a series of areas in which the talks have made progress just in the last three weeks alone, on issues ranging from Cyprus to Gibraltar and security, transport and services relationships, on top of the agreements previously reached.
No 10 sources said her diary would be set out on Monday.
Mrs May will tell the Commons that "taking all of this together, 95% of the withdrawal agreement and its protocols are now settled".
"This week Theresa May will find that she is drinking in the last chance saloon", Andrew Bridgen, a ruling party politician and critic of the Prime Minister, was quoted as saying by The Mail on Sunday.
Some colleagues used much more lurid phrases to describe the Prime Minister's plight.More news: Goodbye Matt LeBlanc! Top Gear announces new presenters
Others spoke of "assassination in the air" and the moment coming "when the knife gets heated, stuck in her front and twisted, she'll be dead soon".
A weekend of feverish speculation suggested Mrs May had 72 hours to save her job ahead of a meeting of the 1922 committee of backbench MPs on Wednesday.
Such language was condemned by some Tory MPs, including Mrs May's critics.
Mrs May has been invited to speak at a meeting of her party politicians on Wednesday and could face a no-confidence motion within days if she fails to convince them she is the best person to lead the Conservatives, according to United Kingdom news media reports on Sunday.
But she will acknowledge the issue of how to deal with the border between Northern Ireland and the Republic of Ireland is continuing to stand in the way of a deal.
Senior Brexiteer Theresa Villiers criticised "disturbing" anonymous briefings to Sunday newspapers, including claims the PM was entering the "killing zone".
However, during exchanges at Prime Minister's Questions, Labour leader Jeremy Corbyn said the Tories were too "weak and divided" to negotiate effectively with Brussels. "It just confirms what numerous people think of us, that we are all potentially out for ourselves and not on the side of working people".More news: New Premier Inn chain shrinks rooms in half
Johnny Mercer, who earlier this week branded the Government a "shitshow", said he had received "overwhelming" support from within the party after speaking out.
Japanese vehicle firms warned that failing to reach a Brexit deal must be "avoided at all costs".
He said: "I think it's understandable there are jitters on all sides of this debate and we need to hold our nerve".
"Now is the time to play for the team".
European Union leaders warned Britain on Thursday they would offer no more concessions to break the deadlock in Brexit negotiations, but expressed confidence that a deal can be done before the country leaves the bloc next March.
With a matter of months to go until Britain leaves the EU, Mrs May is facing mounting pressure from Leavers and Remainers both inside and outside her party.
In a 35-minute conference call with around 130 company bosses, the PM is said to have told them she knows time is running out to reach a Brexit deal, but is confident it can be done.More news: ‘Headless chicken monster’ spotted off east Antarctica for the first time