The main opposition Labour party still opposes the deal, even while May has been wooing individual MPs with promises of protection for workers' rights and new funds for poor towns.
An EU flag flutters during an anti-Brexit demonstration outside the Houses of Parliament in London, Britain January 28, 2019.
"In the event of this vote on Tuesday not going through, nobody knows what would happen".
The mechanism, known as the backstop, is a safeguard that would keep the U.K.in a customs union with the EU until a permanent new trading relationship is in place.
Their talks are focused on the so-called backstop, an arrangement in the Brexit deal meant to keep open the Irish border.
The Telegraph said there was "a backdrop of exasperation" to the Prime Minister's trip to Brussels and that a deal will only succeed if there is "no doubt" that the backstop is temporary.
However, if the deal is passed on Tuesday, the United Kingdom will leave the European Union on 29 March but things will stay broadly as they are until December 2020, while the two sides thrash out a permanent trade deal.
Irish Prime Minister Leo Varadkar said the deal agreed Monday night provided "additional clarity, reassurance and guarantees sought by some to eliminate doubt or fears, however unreal, that the goal was to trap the United Kingdom indefinitely in the backstop".
Lawmakers are expected to try and make changes to the text of the statement to show that there is majority support for an alternative course of action or to make backing the deal conditional on something such as a second referendum.
"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.
"It'll be that little bit easier for the United Kingdom to exit the backstop", she said.
And Mr. Juncker warned Britain "there will be no new negotiations" if lawmakers rejected the deal again.
The European Parliament's chief Brexit official has questioned whether a short extension of the March 29 Brexit deadline can be given if Britain doesn't emerge from its political chaos on the issue.
Residents, businesses and politicians across Britain and the bloc were bracing for a chaotic Brexit after British lawmakers rejected Prime Minister Theresa May's Brexit agreement for a second time by a decisive 391-242 vote on Tuesday.
Last night Mrs May's Brexit deal suffered a fatal blow after Tory Brexiteers and the DUP abandoned her after she failed to secure a breakthrough on the Irish backstop. She has already survived votes of no confidence by her own party and Parliament as a whole.
"Business will continue to leak from London to the European Union, with more activity being booked through local subsidiaries", William Wright, founder and managing director of New Financial told the BBC. After May called a snap election in 2017 and led her party to disastrous results, George Osborne, the former chancellor of the exchequer whom May had sacked, called her a "dead woman walking".
"This evening s agreement with the European Commission does not contain anything approaching the changes Theresa May promised parliament", Labour leader Jeremy Corbyn said.
She warned that voting down "no deal" and asking for extension of the process "does not solve the problems we face".
"The political uncertainty since the referendum and failure to reach a deal has forced firms to prepare for the worst and put their contingency plans into action", says the report.
"It is now for the House of Commons to take an important set of decisions this week", Schinas said.