The government had refused previous requests to publish the advice, which comes just a week before MPs vote on the deal itself, saying it would set a unsafe precedent if the Attorney General could not provide the Prime Minister with honest, confidential legal advice without fear of it being made public.
"I very much understood, however, that many people in this area felt that the impact, both of globalisation in the past 15-20 years and of austerity since 2010, coloured their view of the European Union, as well as concerns about levels of migration".
May said: "If we get to that point there will be a choice between going into the backstop and extending the transition period".
What Theresa May sketched out on Thursday was the idea of allowing MPs to choose when and if they want to go into the controversial "backstop" - the insurance policy against a hard border on the island of Ireland.
Another fellow Cabinet minister, worldwide trade secretary Liam Fox, also backed the deal, warning there was a "natural "Remain" majority" in Parliament and any attempt to overturn the 2016 referendum vote in favour of Brexit would be a "democratic affront".
Mr Cox's advice stated that, if Brussels felt that negotiations on a trade deal had broken down or were taking too long, it would be able to apply to the arbitration panel for Britain to be removed from the customs union while Northern Ireland remains - effectively creating the border in the Irish Sea, which Mrs May has said no prime minister could accept.
The adjustment would mean Parliament would have to approve a decision to trigger the backstop arrangement or extend the transition period beyond December 2020.
He said: "Some people on my side "well they might just vote leave again" and my answers is if they do, they do".More news: Pujara departs for 123 as India end with 250/8
He placed a 10 percent probability on a no-deal Brexit, down from 20 percent, and a 50 percent probability on an orderly Brexit, down from 60 percent.
"The risk is that we end up with no Brexit at all".
May used an interview on BBC radio to press on with her bid to persuade lawmakers to back her deal.
At a hearing in Luxembourg last week, lawyers representing the United Kingdom government and European Union both argued that the case should be rejected.
"I think the Prime Minister has enormous goodwill on both sides of the House. We gave people the choice in the referendum as to whether to leave the European Union or not and they gave us a very clear message". "There are many people who are talking about a second referendum - it isn't anything to do with this deal, it is actually about trying to frustrate Brexit".
He added: "In my view what goes on the ballot is: remain, I hope, with a renewed offer from Europe around issues on immigration and leave, which should be the leave of what I call proper Brexiteers want". "Perhaps the Prime Minister can tell us which paragraphs breach the national interest?"
"All week we have heard from Government ministers that releasing this information could harm the national interest".
Asked if Mrs May still felt she could command a majority in the Commons for the crunch vote next Tuesday, a Downing Street source said: "Everybody knows the parliamentary arithmetic".