Britain's impending split threatened to overshadow an EU summit in the Swedish port city of Gothenburg that was meant to focus on improving social standards and seeing off the threat of populism in the post-Brexit future.
May expressed hopes the bloc would respond "positively" after she met several leaders on the sidelines, but they all warned that time was running out to settle the key divorce issues, and unlock negotiations next month on a trade deal and transition period.
EU Brexit negotiator Michel Barnier warned last week that Britain had just two weeks to meet the bloc's conditions on its divorce bill, citizens' rights and the Irish border if it wanted an agreement.
Irish Prime Minister Leo Varadkar struck a firm line, saying Dublin's demands that Brexit should create no "hard border" between British-ruled Northern Ireland and EU member Ireland must be "written down" in the conclusions of the first phase.
"It's 18 months since the referendum, it's 10 years since people who wanted a referendum started agitating for one, sometimes it doesn't seem like they've thought all this through," he added.
May said Britain would "honour our commitments" on the exit bill, as she promised in a speech in Florence in September, and urged the bloc to start trade talks now.
"I look forward to the European Union responding positively to that so we can move forward together and ensure that we can get the best possible arrangements for the future," May said.
She will also meet European Council President Donald Tusk, with Tusk set to warn her that opening the next phase "is not a given, will require more work and that time is short," an EU source told AFP.
Failure to reach a deal in December would push back a decision until February or March, leaving little time for trade talks before Britain leaves the bloc in March 2019.
British Foreign Secretary Boris Johnson said in Dublin today that doing so would help solve the Irish issue, while Barnier's British negotiating counterpart David Davis called on the EU to compromise across the board.
"Surprise, surprise: nothing comes for nothing in this world," Davis told the BBC in Gothenburg, adding that various EU countries "can see there are big, big benefits in the future deal that we're talking about."
Swedish Prime Minister Stefan Lofven, who also held talks with May, said it was "very dificult to say" whether a deal was possible in December and added that London "needs to clarify what they mean about the financial responsibility."
EU leaders are looking to reboot the union based on plans by France's new president Emmanuel Macron and by Juncker.
"This is absolutely crucial for EU's legitimacy," Lofven told reporters at the meeting, the bloc's first social summit since one in 1997 in Luxembourg.
Most of the EU's 28 national leaders attending with the exception of Chancellor Angela Merkel of Germany, the bloc's economic and political lodestar, who is in Berlin for talks on a new governing coalition.
Lithuanian President Dalia Grybauskaite shrugged off suggestions the Gothenburg summit would be hijacked by the Brexit issue, quipping "No, at least not this one.
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