Asked if there was still enough time left for an orderly Brexit, May said the divorce deal could be fast-tracked through Parliament at the last minute, bypassing usual parliamentary procedure.
The British government classified the meeting as "constructive and Mr Barclay and Mr Barnier agreed to further talks in the coming days and that their teams would continue to work in the meantime on finding a way forward".
As companies and governments across Europe step up preparations for the turmoil of a no-deal exit, diplomats and officials said the United Kingdom now faces three main options: a no-deal exit, a last-minute deal or a delay to Brexit.
As uncertainty around Brexit grows, a study by the IWH Institute in Germany revealed that if Britain leaves the European Union without a deal, it would put almost 6 lakh people around the world out of jobs.
If May fails to secure any changes to the backstop the government will table an amendable motion on 26 February, she said today, with a vote the following day. The free flow of people and goods across the frontier has been an important measure upholding Northern Ireland's peace deal.
Downing Street said Mrs May's statement, which comes a day earlier than expected, will give MPs more time to "digest the content" ahead of a series of expected Commons votes on Thursday.
Among the suggestions is an amendment from Labour MP Richard Godsiff, calling for a second referendum if May is not able to come to a deal backed by parliament.
Mixed bag of weather results in special weather statement for region
Areas north of the King-Snohomish County line, the East Puget Sound Lowlands, and the mountains are likely to see a lot more snow. Another round of winter weather is on the way for Greater Danbury and the Hudson Valley with snow and some mixed precipitation.
Some fear it could leave Britain trapped in the European Union's trade rules with no unilateral way out - but would back a deal if the backstop was time-limited or replaced with alternative arrangements.
But May responded: "In most circumstances, that period may be important in order for this House to have an opportunity to study that agreement". "When we achieve that progress we will bring forward another meaningful vote".
Media captionConfused by Brexit jargon?
Though she did concede she would start to talk to other parties about potential Brexit solutions (what an idea, eh?), May told the Commons in the wake of her victory: "I am pleased that this House has expressed its confidence in the government".
"She is playing for time, and playing with people's jobs, our economic security, and the future of our industries", he said.
With the country's March 29 date for leaving the European Union closing in, Mr Bettel said "a disunited kingdom is more a reality today than a United Kingdom".
As we talked about late on Monday, there has been a sense building in Westminster that the prime minister is, maybe by accident, maybe increasingly by design, looking to nearly the last possible minute for the definitive Brexit vote.
Polar bears invade island town in Russian Federation , locals terrified and emergency declared
But the statement from the town says if there is no way to scare the bears away, killing them will be a last resort. A chain of remote Russian islands has declared a state of emergency after it was swarmed with hungry polar bears .
Speaking to BBC radio Monday morning, Johnson said: "The argument is now about how to get out of the backstop".
Under the suspected plan, May would call time on her Premiership shortly after finally delivering Brexit.
"It is alarming that, despite the overwhelming damage that the Prime Minister's deal will cause to Scotland, the Labour leader would seek to support the deal".
The PM indicated she could well table a deal later than this, saying that the government would, if needed, seek to set aside the usual rules which dictate that worldwide treaties need 21 parliamentary working days before they can be ratified.
"The talks are at a crucial stage", May added.
But in taking the positions that he is taking, Brussels can see that it does not need to move on the deal that Parliament rejected.
United Kingdom and Switzerland strike post-Brexit continuity agreement
But the request was rebuffed by the European Union which has said it will "not reopen the withdrawal agreement". The political wrangling will do little to reassure British business though.