In a statement at Downing Street, the prime minister hit out at the way European Council president Donald Tusk discarded her plans without giving a detailed explanation or offering alternatives.
Ms May said neither of the options previously offered by the EU were acceptable, stated that talks are now at an “impasse” and underlined her willingness to walk away if needs be – finishing with the words “we stand ready”.
But the prime minister also made a new pledge to guarantee the rights of EU citizens living and working in the UK, even in the event of a no-deal Brexit.
Her intervention won approval from cabinet Brexiteers and, to an extent, from MPs on Conservative benches, steadying a volatile political situation which hours earlier had seen critics predicting the collapse of her strategy.
Mr Tusk had surprised British officials by making an unexpectedly strong statement at the end of a summit in Salzburg this week, saying Ms May’s proposals for Brexit “will not work” and following it up with a social media post mocking her negotiating strategy.
With pressure mounting, Ms May acknowledged from inside No 10 that Mr Tusk said the UK proposals would undermine the single market, but added: “He didn’t explain how in any detail or make any counter-proposal. So we are at an impasse.”
She said: “Throughout this process, I have treated the EU with nothing but respect. The UK expects the same. A good relationship at the end of this process depends on it.
“At this late stage in the negotiations, it is not acceptable to simply reject the other side’s proposals without a detailed explanation and counter proposals.
“So we now need to hear from the EU what the real issues are and what their alternative is so that we can discuss them. Until we do, we cannot make progress.”
The prime minister said she would continue preparing a no-deal scenario, but with it looking like a more likely outcome then ever, she went on to make a new commitment to the three million EU citizens currently living in the UK.
She said that even in the event of no deal their rights would be protected, calling them “our friends, our neighbours, our colleagues”.
Ms May also said sought to reassure the people of Northern Ireland that if no deal was possible the government would do “everything in our power to prevent a return to a hard border”.
The Salzburg summit also saw some EU leaders declare the UK should have a second referendum, but Ms May made clear she would not overturn the result of the 2016 vote, claiming that doing so “threatens public trust in our democracy”.
Highlighting her commitment to walk away from talks if the EU fails to engage with her proposals, she said: “I have worked to bring people with me even when that has not always seemed possible.
“No one wants a good deal more than me. But the EU should be clear: I will not overturn the result of the referendum. Nor will I break up my country.
“We need serious engagement on resolving the two big problems in the negotiations. We stand ready.”
European Commission insiders told The Independent that the EU’s negotiators are, and would continue to work constructively towards agreeing a deal with Britain.
But the stern statement from Downing Street came amid the fallout of Ms May’s humiliation in Salzburg, where UK diplomats had hoped to make some progress in bringing EU member state leaders to engage with her plans.
After Mr Tusk said the EU27 were unanimous in their rejection of them, he posted a picture on Instagram of the prime minister and himself at a cake stand with the caption: “A piece of cake, perhaps? Sorry, no cherries.”
It is a reference to a running joke in Brussels that Britain wants to “have its cake and eat it” in talks, while Britain has also been accused of “cherrypicking” because it wants to stay in parts of the single market but not others.
Ms May said in her statement that the EU’s first offer, of staying inside the European Economic Area and customs union, would make a mockery of the referendum result.
She said that the second EU plan, of a trade deal, but with Northern Ireland effectively remaining in the single market and customs union, had already been unanimously rejected by parliament.
The prime minister went on: “It is something I will never agree to – indeed, in my judgement it is something no British prime minister would ever agree to. If the EU believe I will, they are making a fundamental mistake.
“Anything which fails to respect the referendum or which effectively divides our country in two would be a bad deal and I have always said no deal is better than a bad deal.”
She won some support from cabinet Brexiteer Penny Mordaunt, one minister who is rumoured to have been be unhappy wit the PM’s broad approach.
Ms Mordaunt tweeted: “Speaking to constituents today, it is clear that EUs behaviour in recent days is increasing support for us leaving the EU.”
She added: “Ppl still want a deal but content to go without one, even folk who voted remain. Important statement from PM re rights for EU migrants in a no deal scenario.”
Another who is said to have urged the prime minister to move away from her Chequers proposals in recent weeks, Andrea Leadsom, also backed Ms May’s intervention, saying: “The prime minister has negotiated in good faith – respecting the result of the referendum to leave the EU, and keeping the country united.”
Backbench Brexiteer Jacob Rees-Mogg, who chairs the powerful European Research Group, welcomed the prime minister’s “strong and forthright” speech but urged the government to ditch the Chequers plan and come forward with a Canada-style free trade agreement.
He said: “Salzburg was a failure even though European leaders had been briefed about Chequers before it was agreed by the Cabinet.
“It indicates that the EU is not acting in good faith. This makes the prime minister’s task harder and she is right to remind them that no deal is better than a bad deal.”
Labour leader Jeremy Corbyn argued that the prime minister had shown herself to be “incapable of delivering a good Brexit deal”.
He said: “Theresa May’s Brexit negotiating strategy has been a disaster. The Tories have spent more time arguing among themselves than negotiating with the EU.
“From day one, the prime minister has looked incapable of delivering a good Brexit deal for Britain.
“The political games from both the EU and our government need to end because no deal is not an option.”
Carolyn Fairbairn, CBI director-general, urged Brexit negotiators on both sides to change tack.
She said: “The events of the last 24 hours have made one thing abundantly clear – negotiators on both sides must change tack. Rejection of Chequers helps nobody.
“The stakes could not be higher. Jobs, wages and living standards are at risk, on both sides of the Channel.”
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