The British leader has been trying to renegotiate legally binding changes with the EU which would make the deal palatable to Conservative Brexiters from the hardline European Research Group (ERG).
May has spoken by phone to European Commission President Jean-Claude Juncker twice in the past 24 hours.
The EU refuses to budge on the British proposal for what it believes is an attempt to build a unilateral exit mechanism into the Irish backstop, the arrangement that would keep the United Kingdom in a customs union to avoid a hard border on the island of Ireland. As a result, she can not be forced from office for a year.
The House of Commons overwhelmingly rejected the deal in January, primarily because of concerns over arrangements for the Irish border.
Many at Westminster believe that in that event MPs would vote to delay Britain's departure - to the fury of Brexiteers.
"If the votes go this week in a way which means that the prime minister's policy as she has set out and stuck to rigidly over the course of the last two and a bit years is taken away, dismantled slowly by Parliament this week, I think it would be very hard for the prime minister to stay in office for very much longer", Morgan told the BBC.
The former shadow home secretary, who has also been one of the key players in parliamentary moves to avoid no deal, said May should offer a series of indicative votes on the future relationship with Europe and a consultative process with business and trade unions, as well as a possible citizens' assembly.
If the deal is rejected, the most likely next step is that members of Parliament start maneuvering to force the government into a softer Brexit policy that could win cross-party support.
With Britain due to leave the European Union on March 29, and no agreed direction for the world's fifth-largest economy to take, Prime Minister Theresa May will ask lawmakers to vote on up to three possible outcomes.
"We have an opportunity now to leave on March 29 or shortly thereafter and it's important we grasp that opportunity because there is wind in the sails of people trying to stop Brexit", Hunt told the BBC.
Ryanair said a year ago it would have to restrict the rights if United Kingdom shareholders ceased to qualify as European Union nationals to ensure it remained majority EU-owned to comply with its licensing and flight rights.
He and his colleagues insist there is a "silent majority" for a softer form of Brexit, with up to 50 Conservative MPs prepared to back the idea rather than countenance no deal.
May has used various arguments to persuade MPs to back her deal, warning both that defeating it risks a "no deal" Brexit, or that Britain may never leave the European Union at all. That provision will be legally binding, Lidington said. She survived a bid to oust her through a no-confidence vote in December.
Britain's main opposition Labour Party also came out against the agreement.
Ms May arrived in France for Brexit talks with Mr Juncker the day before the British Parliament is set to vote again on a divorce agreement.
She will still hold a second vote on her Brexit deal tomorrow. "Tuesday is about exposing the weakness of the prime minister".