After a dinner of Wiener schnitzel in Salzburg, EU leaders said they will push for a Brexit deal next month but warned May that if she will not give ground on trade and the Irish border by November they are ready to cope with Britain crashing out.
She will then be out of the room when the other 27 leaders discuss her Brexit proposals over lunch, and will find out about their reactions only when summit chair Donald Tusk briefs her separately afterwards.
Shadow Brexit secretary Sir Keir Starmer said: "It has been clear for weeks that Theresa May's Chequers proposals can not deliver the comprehensive plan we need to protect jobs, the economy and avoid a hard border in Northern Ireland".
May said Thursday she would bring forward new ideas on how keep the Northern Irish border open after Brexit, which along with the future economic relationship has proved the key sticking point in negotiations.
She told European Union leaders that their proposal to "carve away" Northern Ireland from the rest of the United Kingdom, to avoid checks on the border after Brexit, was "not credible".
"There is a lot of common ground on the basis of [the United Kingdom plan], especially in the area of domestic security and also foreign cooperation and other issues, but there is still a lot of work to do on the question of how future trade relations will look".
The next major milestone in the Brexit process is fast approaching, with the October 18 summit labelled a "moment of truth" by Mr Tusk.
A Cabinet minister has said there are no changes "on the table at the moment" to Theresa May's Brexit plan, amid blunt warnings from European Union leaders.
Leading Tory Brexiteer Jacob Rees-Mogg welcomed the PM's "strong and forthright" tone, but said it was time for her to ditch Chequers and go for a simple Canada-style free trade agreement with the EU.
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Mrs May addressed EU27 leaders over wiener schnitzel with potatoes and Austrian wine, pitching her Chequers proposal before sit-down meetings begin on Thursday.
The former work and pensions secretary said people like him who sought compromise were left "having more sympathy for the kind of language they [the ERG] have been speaking".
Future UK Prime Ministers must not be allowed to unpickTheresa May's Brexit deal after she leaves No 10, Brussels has warned. Both sides aim to reach a deal that solves the issue.
"We want to avoid a 'No Deal Brexit, ' but we are preparing for that".
Reports suggested a tetchy meeting between May and Irish Prime Minister Leo Varadkar on the sidelines of the summit in Salzburg had helped harden views.
The EU has proposed that Northern Ireland, a British province, continue to follow many EU trade rules and regulations to maintain the status quo with Ireland, a remaining bloc member.
Tusk added he remains "convinced that a compromise, good for all, is still possible" in Brexit negotiations and that he is "a close friend of the United Kingdom and a true admirer of PM May".
In a defiant statement from Downing Street, May blamed Brussels for the "impasse" just weeks ahead of a deadline to seal a deal - and six months before Britain leaves the European Union in March. This would keep Northern Ireland under EU economic oversight if London and Brussels can not agree a trade pact to keep UK-EU borders open after a transition period ends in 2020 - an idea that May and a small party in the province that props up her minority government oppose.
Most EU leaders "would welcome a situation where there is the possibility of the British people putting things into perspective, seeing what has been negotiated, seeing the options, and then deciding once and for all", he added. But she would not say when the proposals would be made public.