European Commission president Jean-Claude Juncker warned Britain would have to leave by May 23 if it did not want to hold elections to the European Parliament - which start on that date - something Mrs May said she is determined to avoid.
The EU has already indicated that it could grant an extension but would only do so if MPs had supported the deal.
But the European Council President said it was "conditional" upon a Theresa May's getting her Brexit deal approved in the House of Commons.
When Mrs May set the March 29 exit date two years ago by serving the formal Article 50 divorce papers, she declared there would be "no turning back", but Parliament's refusal to ratify the withdrawal deal she agreed with the European Union has thrust her government into crisis.More news: Nvidia brings a little RTX DirectX ray tracing to GTX graphics
With fears in Brussels growing that the United Kingdom is heading for a no-deal break, he said he would not hesitate to call an emergency summit next week if that proved necessary.
Some argue that I am making the wrong choice, and I should ask for a longer extension to the end of the year or beyond, to give more time for politicians to argue over the way forward.
The DUP - which is seen as crucial in winning the support of enough Tory MPs to finally pass the withdrawal agreement - has said a deal with the UK Government is still not close to being finalised.
With Mrs May struggling to build a majority to back deal when she brings it back to the Commons for a third time, she made an extraordinary attempt to appeal over the heads of MPs to the public to back her.More news: Smokable Medical Marijuana Now Legal in Florida
In angry exchanges at Prime Minister's Questions, she told MPs that they had indulged themselves on Brexit for too long and that she would never agree to a delay that meant holding European Parliamentary elections.
"Yes, a no-deal Brexit would be damaging and it's absolutely not what we want but I think we're ready to go to this situation because there are only two solutions - either a deal is accepted or there is no deal".
Delaying Britain's departure from the European Union is justifiable if there is a real chance of securing an orderly Brexit, German Foreign Minister Heiko Maas told Germany's Funke media group.
Britain believes it would not have to participate if it got a three-month delay, because the newly elected European parliament is not due to convene until July. On its front page Wednesday, the Brexit-backing Daily Mail newspaper bemoaned the time since the referendum as "1,000 lost days".
As a result, we will now not leave on time with a deal on 29 March.More news: Cindy McCain Shares the Hate That Followed Trump's Criticism
Bloomberg reported it was likely an ultimatum would be issued to Mrs May tomorrow telling her that she has only a matter of weeks to decide if she feels she can get the deal through Parliament.
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