The Prime Minister will meet the executive of the 1922 Committee to set out plans for her departure, with another ultimatum set to be delivered by Tory backbenchers.
Brexit Secretary Steve Barclay conceded it would be the end of the road for the deal if the Withdrawal Agreement Bill (WAB) is defeated when it is put before the Commons in June.
Downing Street sources insisted another loss would not represent a vote of no confidence in the Prime Minister, but said the significance “won’t be underestimated”.
Mr Barclay said the WAB would be published as soon as possible and would have its second reading in the Commons, the first legislative hurdle it would have to clear, in the week beginning 3 June.
He told the Lords EU Committee: “I think if the House of Commons does not approve the WAB then the [Michel] Barnier deal is dead in that form.”
Number 10 said efforts were continuing to find a “sustainable” majority for the deal, with discussions involving DUP and Labour MPs.
However, the DUP’s Westminster leader Nigel Dodds said that unless Mrs May can “demonstrate something new that addresses the problem of the backstop then it is highly likely her deal will go down to defeat once again”.
Both Mr Barclay and International Trade Secretary Liam Fox warned Eurosceptics that the possibility of Brexit not happening at all would increase if the WAB was defeated.
That would leave a no-deal Brexit, which has previously been rejected by MPs, or the revocation of Article 50, cancelling the entire process, as the only options when the 31 October deadline arrives.
Cross-party Brexit talks between Labour and the Tories have failed to produce a compromise, with Jeremy Corbyn’s spokesman saying the party would not support the WAB if no agreement was reached.
The spokesman said that unless there was an agreement based on “real compromise and movement by the Government” then the WAB would be “based on the same botched Brexit deal that has been rejected three times already by Parliament”.
At Prime Minister’s Questions, Mrs May was challenged by Tory Eurosceptic Peter Bone, who said activists in his constituency favoured a no-deal Brexit and wanted her to quit before the May 23 European elections.
“They say that her deal is worse than staying in the European Union, and that they want us to come out now on a no-deal basis,” Mr Bone said, adding that Tory members had “lost confidence in the Prime Minister”.
He said afterwards: “The definition of madness to keep repeating the same thing and expecting to have a different result.”
The SNP’s Westminster leader said it was “complete unacceptable” for the government to force through a “shoddy backroom deal” with Labour on Brexit.
“Scotland did not vote for Brexit and we should not be dragged out of the EU against our will,” he said. “At the European elections next Thursday, the people of Scotland will make their voices heard – whether Westminster likes it or not.”