May has asked EU leaders for a three-month delay to Brexit to allow her to try one more time to get her withdrawal agreement through parliament, and the bloc has indicated it could postpone exit day to May 22.
But as they arrived at a two-day summit in Brussels dominated by Britain's deep political crisis, the other 27 leaders warned that this would be possible only if MPs approve next week the deal they have already resoundingly rejected twice.
"In the case of a negative British vote then we'd be heading to a no deal," French President Emmanuel Macron told reporters.
"We all know it. And it's essential to be clear in these days and moments."
'Everything will be more difficult'
Dutch Prime Minister Mark Rutte said he expected his fellow European leaders to agree in principle on Thursday to May's request for a delay.
But they would then "put it fairly and squarely at the door of the British parliament to make that call", adding: "In case they say no, I'm not going to speculate."
European Commission President Jean-Claude Juncker said that May "has to make sure that the British parliament agrees to the withdrawal deal".
If that does not happen early next week, "everything will be more difficult", he said, adding that EU leaders would have to meet again before Brexit day -- currently just eight days away on March 29.
German Chancellor Angela Merkel added: "We should do everything we can until the last moment to make an orderly Brexit possible."
The embattled May also refused to rule out the possibility that Britain severs its 46-year relationship with the EU next week with no new arrangements in place.
"What matters is that we recognise that Brexit is the decision of the British people and we need to deliver on that," she said as she arrived in Brussels.
She added: "I sincerely hope we can do that with a deal."
A "no deal" next week risks huge economic disruption on both sides of the Channel.
In a joint open letter to May on Thursday, British business and union leaders warned: "Our country is facing a national emergency."
May wrote to EU Council president and summit host Donald Tusk on Wednesday to ask to delay Brexit until June 30, to ratify her deal if it is agreed by parliament.
But EU diplomats said the consensus was that Britain would either have to leave before the May 23 European election or take part in the vote and stay in the bloc until at least the end of the year.
May has insisted that she will not oversee a long delay, which would increase pressure at home for a general election or for a second Brexit referendum that might reverse the decision to leave.
She repeated on Thursday that requesting even a short delay was a matter of "personal regret".
She continues to work to persuade her MPs to support the deal, but appeared to have undermined that effort with a speech to the nation on Wednesday when she blamed parliament for the Brexit deadlock.
"All MPs have been willing to say is what they do not want," she complained, in a televised address that infuriated lawmakers on all sides.
Tusk admitted that European leaders are suffering "Brexit fatigue".
He said he feared his hopes for an orderly Brexit agreement with the agreed plan in place may prove "frail, even illusory."
British lawmakers have twice resoundingly rejected May's agreement, and a third vote the premier hoped to hold this week was cancelled by the parliament speaker using a 400-year-old precedent.
Nevertheless, Tusk was clear that Europe wants May to try again, saying "a short extension will be possible but it will be conditional on a positive vote on the withdrawal agreement in the House of Commons".
The British parliament has been deadlocked for months over Brexit, with MPs unable to decide how to implement the result of the 2016 referendum, and voters themselves are also sharply divided.
As May and EU leaders gathered Brussels, opposition Labour leader Jeremy Corbyn was holding his own talks in the Belgian capital, including with the EU's Brexit negotiator, Michel Barnier.
"We are looking for an alternative that can command a majority in parliament... that can bring about a resolution of this crisis," Corbyn said.