Scrambling to plot an orderly path out of the Brexit maze just days before the United Kingdom is due to leave on March 29, May rushed to Strasbourg to agree additional assurances with European Commission President Jean-Claude Juncker.
On 30 January MPs voted for an amendment by 317 votes to 301 to pass the Brexit deal bill if Theresa May could negotiate changes to the Irish backstop. The newspaper also said cabinet ministers had spoken about whether to insist she goes as early as this week.
"Tomorrow the House of Commons will debate the improved deal that these legal changes have created".
In particular, the lawmakers advised the prime minister in phone conversations that she should delay the vote and put forward a motion describing a divorce agreement that would finally heal divisions over the issue within May's own party, the media outlet said.
"We are in very perilous waters, and people who want to make sure that we really do deliver this result need to remember that if it fails. they are going to say: "There was a party that promised to deliver Brexit, we put them into No 10 and they failed", and the consequences for us as a party, would be devastating", he said.
BBC Brussels reporter Adam Fleming said the mood was "bleak" in Brussels after the EU's chief Brexit negotiator, Michel Barnier, briefed European Union ambassadors on the state of play earlier.
"And the deal sets us on course for a good future relationship with our friends and allies in the EU".
Britain's official deadline to leave the European Union is on March 29.
All EU member states have to agree to an extension, and some EU leaders have expressed concern that it needs to have a specific objective.
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The EU is preparing for a delay to Brexit of at least a year, senior officials and diplomats have said.
Due to the sensitive nature of the region and because the free passage is one of the crucial articles of the 1998 Belfast Agreement, the U.K., EU, Northern Ireland and Ireland all reject the idea of returning to a hard border where checkpoints and customs buildings will need to be installed.
Some members of Parliament are telling May to pull the vote because they expect a defeat on a similar scale to the historic defeat she suffered in January, the Times reported.
"The EU will continue working intensively over the coming days to ensure that the United Kingdom leaves the EU with an agreement".
Talks in Brussels have been deadlocked and over the weekend, it became clear that May was unable to convince European Union leaders to agree to changes she was seeking to the withdrawal agreement, particularly on the so-called "backstop", the protocol to ensure an open border is maintained in the island of Ireland.
This vote is likely to see no deal taken out of the Brexit equation - leading to a third vote on Thursday for Brexit to be delayed altogether.
What happens if Parliament votes in favor of May's deal? She survived a bid to oust her through a no-confidence vote in December, so can't be forced from office for a year.
Alan Wager, a Brexit expert at the U.K.in a Changing Europe think tank, said Parliament this week could decisively rule out both May's deal and a no-deal departure.
There has been speculation that the beleaguered prime minister may be forced to resign.