Parliament has rejected the agreement twice, but May aims to try a third time this week if she can persuade enough lawmakers to change their minds. Her aim is to have the deal agreed before EU leaders meet Thursday for a summit in Brussels.
May's goal is to win over Northern Ireland's small, power-brokering Democratic Unionist Party. The DUP's 10 lawmakers prop up May's Conservative government, and their support could influence pro-Brexit Conservatives to drop their opposition to the deal.
Still, May faces a struggle to reverse the huge margins of defeat for the agreement in Parliament. It was rejected by 230 votes in January and by 149 votes last week.
Influential Conservative Brexiteer Jacob Rees-Mogg said he would wait to see what the DUP decided before making up his mind on whether to support May's deal.
"No deal is better than a bad deal, but a bad deal is better than remaining in the European Union," he told LBC radio.
British Foreign Secretary Jeremy Hunt said Monday he saw "cautious signs of encouragement" that the deal might make it through Parliament this week.
After months of political deadlock, British lawmakers voted last week to seek to postpone Brexit. That will likely avert a chaotic British withdrawal on the scheduled exit date of March 29 -- although the power to approve or reject a Brexit extension lies with the EU, whose leaders are fed up with British prevarication.
May is likely to ask for an extension at the Brussels summit, and EU leaders say they will only grant it if Britain has a solid plan for what to do with the extra time.
"We have to know what the British want: How long, what is the reason supposed to be, how it should go, what is actually the aim of the extension?" German Foreign Minister Heiko Maas told reporters in Brussels. "The longer it is delayed, the more difficult it will certainly be."
Belgian Foreign Minister Didier Reynders agreed, saying: "We are not against an extension in Belgium, but the problem is -- to do what?"
Opposition to May's deal centers on a measure designed to ensure there is no hard border between the U.K.'s Northern Ireland and EU member Ireland after Brexit.
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. Brexit supporters in Britain fear the backstop could be used to bind the country to EU regulations indefinitely, and the DUP fears it could lead to a weakening of the bonds between Northern Ireland and the rest of the U.K.
Talks between the government and the DUP are aimed at reassuring the party that Britain could not be trapped in the backstop indefinitely.
May has said that if her deal is approved by Parliament this week, she will ask the EU to delay Brexit until June 30 so that Parliament has time to approve the necessary legislation. If it isn't, she will have to seek a longer extension that could mean Britain participating in May 23-26 elections for the European Parliament.
May said in an article for the Sunday Telegraph that failure to approve the deal meant "we will not leave the EU for many months, if ever."
"The idea of the British people going to the polls to elect MEPs (Members of the European Parliament) three years after voting to leave the EU hardly bears thinking about," she wrote.
But May suffered a setback Monday when former Foreign Secretary Boris Johnson refused to support her deal.
Johnson, a staunch Brexiteer, used his column in the Daily Telegraph to argue that the backstop left the U.K. vulnerable to "an indefinite means of blackmail" by Brussels.