In the past few hours Mrs May held a late-night meeting with the European Commission President Jean-Claude Juncker in Strasbourg to secure the agreement.
The House of Commons overwhelmingly rejected the deal in January, primarily because of concerns over arrangements for the Irish border.
And Tory Brexiteer Peter Bone urged the government to delay the vote until MPs had had enough time to scrutinise any changes to the deal.
Irish Foreign Minister Simon Coveney told reporters in Dublin that May would travel to Strasbourg - where the European Parliament is holding its plenary session this week - "to finalize an agreement, if that's possible".
That's unlikely. There are a few hard-core "Brexiteer" members of Parliament in May's Conservative Party who support leaving the European Union with no deal, and a recent poll showed a majority of the party's rank-and-file members agree.
"It would be very hard for the prime minister to stay in office for very much longer", Morgan told the BBC. "There will be no third chance, there will be no further reassurance on the assurances if this deal fails", Juncker said.
All EU member states have to agree to an extension, and some EU leaders have expressed concern that it needs to have a specific goal.
"Let's be crystal clear about the choice: it is this deal or Brexit might not happen at all", he said.
But eurosceptic conservative MPs, as well as the Democratic Unionist Party (DUP), May's ally in Northern Ireland, have been opposed to it arguing it would tie the United Kingdom to the EU's trade rules indefinitely, or create a border between Northern Ireland and the rest of the UK.More news: R Kelly released from jail after making child support payment
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The PM was expected to give a statement following talks but is still "in negotiations", her deputy David Lidington said.
This would keep Britain in the EU's customs union and parts of its single market until and unless another way - such as a trade deal - is found to avoid frontier checks.
But as MPs prepare to vote again on the deal on Tuesday, the prime minister has little to show for her efforts, prompting warnings of another humiliating defeat.
"Now is the time to come together to back this improved Brexit deal and deliver on the instruction of the British people", she said.
The prime minister has warned that a defeat risks scuttling her blueprint for Brexit and replacing it with a scaled-down plan, potentially keeping the United Kingdom inside the EU's single market and customs union.
The government has previously tried to use the risk of Brexit being reversed as a way to convince eurosceptics to back May's deal despite their deep reservations about it.
With a number of Labour MPs against a second vote, it remains unclear whether it would find a majority in Parliament.
May offered lawmakers a "meaningful" vote on what she had hoped would be a revised deal on Tuesday but with no major changes yet secured, Brexit-supporting lawmakers warned it would be defeated again.
But MPs rejected it in January by a massive 432 votes to 202, with many of May's Conservatives rebelling against her. "Tuesday is about exposing the weakness of the prime minister".