Lawmakers are preparing to debate proposals for a series of votes on alternatives to May's deal, which could take control of the process away from the government.
The European Commission has warned that is it now "increasingly likely" that the United Kingdom will leave the EU without a deal in two-and-a-half weeks time.
May too has made clear that she would not implement a proposal that ran counter to her election manifesto, which promised a clean break with the EU.
The ideal scenario is May "gets her deal over the line first, and then announces that she's resigning so that there can be an orderly replacement of the prime minister", he said.
The amendment A has been proposed by a cross-party group of lawmakers, led by Oliver Letwin, a member of May's Conservative Party.
However, her resignation would start a Conservative leadership contest, with the victor automatically becoming Prime Minister. Thursday would be the most likely day.
Some of her lawmakers have already publicly urged her to go.
It is not yet clear whether this proposal - which would involve accepting the free movement of workers from other European Union countries - would command the support of a majority within Parliament.
These opponents of May's deal think London's best bet is to make a clean break with Brussels without any binding deal, allowing Britain to immediately set lower tariffs for trade outside the European Union and finalise deals with the rest of the world.
Brexit Secretary Steve Barclay said an election could be the outcome if lawmakers back proposals contrary to the pledges on which the government was elected.
Her hopes of winning DUP support for the Brexit deal also suffered a setback after a phone call with Arlene Foster failed to change the unionist party's opposition to the Withdrawal Agreement.
The leaders rejected May's request to extend the deadline until June 30.
Having defeated a motion of no-confidence in January, Mrs May is immune from a leadership challenge for a year.
May's spokesman, James Slack, says the prime minister is determined to win support for her deal, but "we will only bring the vote back if we believe we would be in a position to win it".
At a meeting Sunday at the prime minister's country retreat, Chequers, prominent Brexiteers told May they might back the deal - if she agreed to step down so that a new leader could take charge of the next phase of negotiations, which will settle Britain's future relations with the EU.
Now a May 22 departure date will apply if parliament rallies behind the British prime minister and she is able to pass her deal.
A video grab from footage broadcast by the UK Parliament´s Parliamentary Recording Unit (PRU) shows Britain´s Prime Minister Theresa May listening as MPs speak in the House of Commons in London on March 25, 2019.
Earlier, May's divided cabinet of senior ministers had met to discuss a way forward.
Britain's bickering parliament has splintered into rival factions over the direction of Brexit, cutting across traditional party loyalties and forging odd alliances.
Gove also downplayed the possibility of ousting May: "I think it is not the time to change the captain of the ship".