Parliament must approve the deal before it can be ratified.
Earlier, Mr Barnier said that talks about the UK's withdrawal from the bloc were now between the British government and MPs.
"There will be no third chance".
"MPs were clear that legal changes were needed to the backstop".
"Today we have secured legal changes", May said in a late night news conference in Strasbourg beside Juncker, exactly 17 days before the United Kingdom is due to leave the EU.
As the British Parliament still has not cleared the Brexit deal negotiated between the United Kingdom government and the European Union, there is a real risk of the United Kingdom crashing out of the European Union without the necessary arrangements for airlines.
Cooper had said she and Conservative MPs would attempt to force votes on an extension of article 50 on Thursday should the prime minister pull the votes, saying it would be a "straight-up lie" to parliament if May changed her plans.
Alan Wager, a Brexit expert at the U.K.in a Changing Europe think tank, said Parliament this week could decisively rule out both May's deal and a no-deal departure.
"We are committed to ratifying this deal before 29 March", the spokesperson said, adding that it was now for Britain's MPs to set the course for the next steps for Brexit.
Cabinet ministers urged their Tory colleagues on Sunday to support May's deal, with Foreign Secretary Jeremy Hunt saying both the European Union and British parliamentarians need to be "realistic". We want to keep our journalism open and accessible and be able to keep providing you with news and analysis from the frontlines of Israel, the Middle East and the Jewish World.
The DUP's support is crucial if the deal is to pass the House of Commons.
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But MPs on all sides in London were swift to condemn it for a variety of reasons, and it was rejected in January by 432 votes to 202. "If she won't find a way forward, Parliament has a responsibility to do so instead", Cooper said.
They fear the deal will leave Britain a "vassal state" with its plans for a transition period and the "backstop" plan for the Irish border, and chafe at the £39 billion exit bill. She has already survived votes of no confidence by her own party and Parliament as a whole.
The Commons later sent her back to renegotiate the backstop.
The airline is to put plans in place to protect themselves from a no-deal and wants to ensure they will remain a majority European Union controlled and owned airline.
If her deal is defeated, May has said she will give lawmakers a vote on Wednesday on leaving without a deal on March 29.
Some members of Parliament are telling May to pull the vote because they expect a defeat on a similar scale to the historic defeat she suffered in January, the Times reported.
Any postponement would have to be approved by the leaders of the other 27 nations, who are next meeting at a Brussel's summit on March 21 and 22 - a week before Brexit day.
Pro-Brexit U.K. lawmakers said they would read the fine print and wait for the judgment of Britain's attorney general before deciding how to vote on Tuesday.
"This is all words and twisted meanings".
Britain's main opposition Labour Party also came out against the agreement.