If the vote on Tuesday fails, May has pledged to give the House of Commons an opportunity to vote on two additional resolutions: One on a no-deal Brexit; the other on delaying Brexit beyond the March 29 date enshrined in law.
In a dramatic late announcement to the House of Commons on Monday night, Lidington said May had struck the agreement after an 11th-hour dash to Strasbourg to meet European Union leaders, raising her hopes of persuading Eurosceptic MPs to back her in the House of Commons on Tuesday.
Commenting on the Strasbourg agreements, the Attorney General wrote: "I now consider that the legally binding provisions of the Joint Instrument and the content of the Unilateral Declaration reduce the risk that the United Kingdom could be indefinitely and involuntarily detained within the Protocol's provisions at least in so far as that situation had been brought about by the bad faith or want of best endeavours of the EU".
Brexit-supporting MPs had said they would look at what Mrs May achieved before the vote, but that she would have to show a clear way for the backstop to end.
May said documents to be added to the deal provided "legally binding" assurances that the backstop would be temporary and that Britain would have a way to get out of it if the European Union failed to negotiate in good faith. A version of the deal was rejected in parliament in late January by a record 230-vote majority against a government proposal. It has the same legal status as the Withdrawal Agreement.
Steve Baker, a leading figure in the ERG, said the government had put "a very good gloss on something that falls short".
The shadow Brexit secretary, Keir Starmer, immediately cast doubt on whether the changes would actually be legally binding, saying: "It sounds again that nothing has changed".
If she loses the vote, she has said MPs will get a vote on Wednesday on whether to leave without a deal and, if they reject that, then a vote on whether to ask for a limited delay to Brexit.
However-and this is a huge "however"-a rejection of the concept does not mean a no-deal Brexit is off the table".
"Whenever there is any update to Brexit negotiations or parliamentary votes, we see a spike in support requests as customers try to understand what it all means for the pound and our Brexit Support Desk will be anticipating lots of queries today and tomorrow once the results are known".
If they vote yes to extend Article 50 then the United Kingdom will ask for a short extension. Even if he changes his legal advice and endorses the new deal, May still has a huge task ahead to persuade enough members of her own Conservative Party to support her plan in Tuesday night's vote.
And if that is turned down MPs will have a third vote in three days on whether to extend Article 50 and delay Brexit.
"This evening's agreement with the European Commission does not contain anything approaching the changes Theresa May promised parliament", said Labour leader Jeremy Corbyn.