In a speech repeatedly interrupted by MPs attacking her deal, the prime minister pledged to give Parliament and the devolved administrations a "greater and more formal role" in forthcoming negotiations with the European Union over trade - but declined to say whether MPs would get a vote on that deal.
She suggested MPs could be "given a role" in deciding whether to activate the backstop, which is created to stop the return of a physical border.
The Home office published plans on Thursday which revealed that the policy for those from other European Union countries living in the United Kingdom will shift slightly to make deadlines tighter and making it harder for families to move over.
Mrs May faced calls to postpone Tuesday's vote with senior Conservative MP Sir Graham Brady, chairman of the backbench 1922 Committee, saying he would welcome the vote being deferred if no solution could be found to differences within the party over the backstop.
Blair, who won three general election victories as PM, said a second referendum should settle the argument, 30 months after the leave vote.
It is the arrangement included in the withdrawal agreement to ensure there is no return to a hard border - physical checks or infrastructure - between Northern Ireland and the Republic of Ireland after Brexit.
Led by Hugo Swire, the alternative plan would give MPs the chance to vote to extend the transition period at the end of 2022 - rather than falling into the backstop.
Speaking recently, he said: "I'm absolutely diametrically opposed to it and I'm assured by a very large number of colleagues that they hold a similar opinion".
"There are questions about how decisions are taken as to whether we go into the backstop, because that isn't an automatic", May said.
Shadow Brexit secretary Sir Keir Starmer said that the 33-paragraph document revealed "the central weaknesses in the Government's deal".
"This risk must be weighed against the political and economic imperative on both sides to reach an agreement that constitutes a politically stable and permanent basis for their future relationship.
Perhaps the Prime Minister can tell us which paragraphs breach the national interest?"
In dramatic scenes at Westminster, the government bowed to pressure to publish the "final and full" legal advice to Cabinet on the deal after MPs voted by 311 to 293 that its failure to do so amounted to contempt.
"I absolutely agree that it is the duty, I believe, of this Parliament, it is the duty of us as politicians to deliver on the result of the vote that the British people gave in 2016 in the referendum".
This was echoed by Mr Hammond who has told MPs it is "simply a delusion" to think a better Brexit deal can be renegotiated at the 11th hour after warning a no-deal is "too terrible to contemplate".
The Democratic Unionist Party (DUP), which normally votes with the government under its confidence-and-supply pact, was among parties that voted to force the publication of the legal advice.
On Wednesday, May's top parliamentary enforcer, or chief whip, Julian Smith, spent an hour meeting with pro-Brexit Conservative and DUP lawmakers, listening to their concerns about the deal.
Labour leader Jeremy Corbyn said Mrs May's deal would "make this country worse off".
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