Britain leaves the EU on March 29 — the first country ever to do so — but a deal must be sealed in the coming weeks to leave enough time for the U.K. and European Parliaments to sign off. May faces increasing domestic pressure over her proposals for an agreement following the resignation of another government minister last week.
The British leader had been hoping to present a draft deal to her Cabinet this week. But no Brexit breakthrough was announced Monday after talks between European affairs ministers. The two sides are locked in technical negotiations to try to bridge the final gaps in a move laden with heavy political and economic consequences.
May said talks were in their "endgame" but that negotiating a divorce agreement after more than four decades of British EU membership was "immensely difficult."
May told an audience at the Lord Mayor's Banquet in London that "we are working extremely hard, through the night, to make progress on the remaining issues in the Withdrawal Agreement, which are significant.
"Both sides want to reach an agreement," May said, though she added she wouldn't sign up to "agreement at any cost."
The main obstacle to a deal is how to keep goods flowing smoothly across the border between EU country Ireland and Northern Ireland in the U.K.
Both sides have committed to avoid a hard border with costly and time-consuming checks that would hamper business. Any new customs posts on the border could also re-ignite lingering sectarian tensions. But Britain and the EU haven't agreed on how to achieve that goal.
"Clearly this is a very important week for Brexit negotiations," Irish Foreign Minister Simon Coveney told reporters after the meeting in Brussels. "The two negotiating teams have really intensified their engagement ... There is still clearly work to do."
And Martin Callanan, a minister in Britain's Brexit department, said all involved were "straining every sinew to make sure that we get a deal but we have to get a deal that is right for the U.K., right for the EU and one that would be acceptable to the U.K. Parliament."
EU Brexit negotiator Michel Barnier didn't speak to reporters Monday and a planned news conference with him was canceled.
Instead, EU headquarters issued a short statement saying that Barnier explained to the ministers that "intense negotiating efforts continue, but an agreement has not been reached yet."
Dutch Foreign Minister Stef Blok said the two sides "are getting closer to each other."
"But in negotiations there is only a deal if there is full agreement," Blok said. "There is only a 100-percent deal. There is not a 90-percent deal or a 95-percent deal."
Earlier, France's EU affairs minister, Nathalie Loiseau, stepped up pressure on May. "The ball is in the British court. It is a question of a British political decision," she said.
The EU is awaiting Barnier's signal as to whether sufficient progress has been made to call an EU summit to seal a deal.
Rumors have swirled of a possible top-level meeting at the end of November. But Austrian EU affairs minister Gernot Bluemel, whose country holds the EU's rotating presidency, said "so far progress is not sufficient to call in and set up another (summit)."
In recent days there have been signs of progress behind the scenes, but all parties have remained tight-lipped about the developments, given the politically charged atmosphere.
In Britain, pro-Brexit and pro-EU politicians alike warned May that the deal she seeks is likely to be shot down by Parliament.
Boris Johnson, a staunch Brexit supporter, wrote in a column for Monday's Daily Telegraph that May's plan to adhere closely to EU regulations in return for a trade deal and an open Irish border amounts to "total surrender" to the bloc.
The proposed terms are scarcely more popular with advocates of continued EU membership.
Former Education Secretary Justine Greening on Monday called May's proposals the "worst of all worlds," and said the public should be allowed to vote on Britain's departure again.
"We should be planning as to how we can put this final say on Brexit in the hands of the British people," Greening told the BBC.
Johnson's younger brother, Jo Johnson, resigned last week backing calls for a second referendum on whether the country should leave the EU. May has consistently rejected the idea of another nationwide vote on Brexit.