If the government's deal is not passed on Tuesday, a vote on a no-deal scenario is expected to take place on Wednesday.
"The probability of some variant of this deal going through is falling rapidly and the probability of an alternative, nearly certainly softer, deal replacing it is rising", said Adam Cole, chief currency strategist at Royal Bank of Canada.
Labour Party leader Jeremy Corbyn urged MPs to vote against the deal when it returns to the Commons, saying May had "recklessly run down the clock" on negotiations.
The second vote on May's Brexit deal, up until last night, was widely believed to be likely rejected.
The Conservatives' partner in Parliament, the DUP, have been outspoken critics of the deal - especially the backstop.
What happens if MPs back the delay?
In essence, the assurances give the United Kingdom a possible path out of the backstop through arbitration and underscore the EU's repeated pledges that it does not want to trap the United Kingdom in the backstop. It could lead to no deal at a later date if no other outcomes can be found.
Around 320 votes are needed to guarantee a majority. Would the three months mentioned by Theresa May be long enough? The EU would wish to avoid the problems this threw up.
Before the latest Brexit developments, risk sentiment was already well supported after global equity indexes climbed overnight on gains in technology stocks and expectations of more stimulus from China.
It remains to be seen whether the changes announced will reassure those MPs, who had wanted to see either a time limit on the backstop, or a unilateral exit clause for the UK.
That, in turn, would make such options as a new Brexit referendum or a "softer" withdrawal from the European Union lot more likely, he said.
"The Prime Minister and I have agreed on a joint legally binding instrument relating to the Withdrawal Agreement".
They have been demanding that the agreement does not keep the United Kingdom locked in the Irish backstop - an insurance policy to guarantee no return to a hard border in Ireland - indefinitely. The U.K. would enter a transition period that would last through the end of 2020.
The Independent Group - the recently-formed contingent of 11 former Labour and Tory MPs - has tabled an amendment to the motion to be debated, calling for an extension to Article 50 so Parliament can agree the terms of a so-called People's Vote.
First is a "joint legally binding instrument" on the withdrawal text, which addresses the deal's controversial "backstop" plan to keep open the border between European Union member Ireland and the British province of Northern Ireland.
The changes agreed on Monday came in the form of two documents.
He described the texts as a "legally binding instrument on the Withdrawal Agreement and protocol on Northern Ireland and a joint statement to supplement the political declaration".
"I accept the point that if both sides are negotiating in good faith, that is what the final paragraph (of the Attorney General's advice) is referring to and therefore there will need to be an arrangement".
"There will be a specific negotiating track on alternative arrangements from the very start of the next phase of negotiations", May promised.