The Prime Minister said Britain could be forced to leave the EU without a deal if they did not give ground.
Mrs May issued her dramatic ultimatum at a private meeting with German chancellor Angela Merkel, French president Emmanuel Macron, EU commission president Jean-Claude Juncker, EU Council president Donald Tusk and Dutch prime minister Mark Rutte.
The Prime Minister said Britain could be forced to leave the EU without a deal if they did not give ground
She said that without the promise of help she would put her Brexit deal to a vote in the Commons next week.
With more than 100 Tory MPs and her DUP partners currently opposed to it, the deal faced certain defeat on a scale likely to kill it off forever.
Downing Street last night refused to confirm that Mrs May had threatened to crash her own deal.
But sources said she was preparing to bring back the vote to the Commons next week if they had continued their hardline stance.
Last night the gamble appeared to have paid off, with Mrs May telling reporters she was now convinced the EU was willing to budge on the Irish backstop, which critics claim could leave the UK trapped in a customs union against its will.
It came after a stormy Brussels summit at which EU leaders stonewalled on help to save the PM’s Brexit deal – and she rounded on Mr Juncker for apparently describing her demands as ‘nebulous and imprecise’.
Downing Street last night refused to confirm that Mrs May had threatened to crash her own deal
Mrs May had appeared to come away virtually empty-handed from the summit. But, speaking at a defiant press conference, she insisted she could get the refinements needed in time for a vote on her deal, which she has promised by January 21.
‘I never said it was going to be easy,’ she added.
Speaking after her private meeting yesterday with the EU chiefs, the PM said: ‘My discussions with colleagues today have shown that further clarification and discussion following the Council’s conclusions is in fact possible.
‘There is work still to do and we will be holding talks in coming days about how to obtain the further assurances that the UK Parliament needs in order to be able to approve the deal.’
The Irish backstop is designed to prevent the emergence of a hard border on the island of Ireland if trade talks falter.
Mrs May had appeared to come away virtually empty-handed from the summit
But critics fear it could leave the UK trapped in a customs union against its will for years.
It is understood Mrs May asked for assurances on a future trade deal – which would negate the need for the backstop – to be added to the withdrawal agreement.
Mrs May was seeking assurances with ‘legal force’ that the backstop could only be ‘temporary’.
EU leaders were supposed to sign up to a series of summit conclusions declaring that the backstop ‘does not represent a desirable outcome for the EU’. A proposed phrasing also suggested that if the backstop were used it would only apply for a ‘short period’.
But both of these phrases were stripped out of the final conclusions following a row over dinner between the other 27 leaders.