A visibly angry Theresa May desperately sought to keep alive her beleaguered Brexit plan after EU leaders tried to kill it off at the Salzburg summit with Donald Tusk saying bluntly it “will not work”.

After a “frank” exchange with the European Council President behind closed doors, an exasperated Prime Minister addressed a press conference where she insisted her Brexit blueprint was the “only serious and credible proposition on the table” but admitted there remained "a lot of hard work to be done" to secure agreement with just four weeks to go to the October EU summit.

Yet she stressed: “If the political will is there on the other side, I'm confident we will reach a deal and to do so is in the EU's interests as well as the UK's."

However, the EU27’s rejection of her Brexit blueprint means that Mrs May faces a torrid time at the intervening Conservative conference in Birmingham when the Brexiteers, led by Boris Johnson, will seek at every turn to undermine her compromise plan.

David Davis, the former Brexit Secretary, said: “It’s time for a reset; it’s time for a rethink.”

Nigel Farage, the former Ukip leader, said the EU's "gangsters and bullyboys" would never be satisfied and urged the PM to pursue a "simple free trade deal".

In Salzburg, Mrs May confirmed that she would bring forward a “new proposal” to try to unblock the logjam on the crunch issue of the Irish border.

It is thought this would entail having “regulatory checks” at Irish Sea ports; these are technically different but similar to customs checks. The key will be not just whether Brussels can accept the proposal but whether the Democratic Unionists, who provide the PM with her Commons majority, will be able to as well.

Despite the informal Austrian summit being trailed as an occasion where Mrs May’s fellow EU leaders would seek to be helpful and sympathetic to her embattled position, a Whitehall source said a clash was “inevitable” but stressed the PM had shown she would “not just roll over”.

After the PM gave her 10-minute Chequers Plan presentation over dinner the night before, Mr Tusk said: "Everybody shared the view that while there are positive elements in the Chequers proposal, the suggested framework for economic co-operation will not work; not least because it risks undermining the single market."

Angela Merkel gave a downbeat assessment, saying "substantial progress" was needed on the UK's withdrawal agreement by the European Council on October 18/19 and that there was "still a large piece of work" to be done on the separate issue of future trade relations with the UK.

The German Chancellor stressed how the EU27 was "united” and, on the matter of the single market, there could be “no compromises”.

She declared: “No-one can belong to the single market if they are not part of the single market."

But Mrs Merkel also made clear that, with creativity, a "practical, good, close" future relationship with the UK could still be delivered.

Emmanuel Macron, the French President, launched a stinging attack on Britain’s Brexiteers, saying: “With Brexit, and I say it with the utmost respect for the British people, we have demonstrated that those who said they could do easily without Europe, that everything is going to be alright, are liars.”

Viktor Orban, the controversial Hungarian leader, who is at odds with Brussels over issues such as human rights and the rule of law, stressed that he was against any approach to punish the British just because they had decided to leave.

"It's a great nation so we should have a fair approach to have a good deal,” he added.

At home, Labour’s Sir Keir Starmer, said it had been clear for weeks that the Chequers Plan could not deliver the comprehensive plan needed to protect jobs, the economy and avoid a hard border in Northern Ireland.

"With just weeks to go until a deal must be struck, the Prime Minister cannot keep ignoring this reality. She needs to urgently drop her reckless red lines and put forward a credible plan for Brexit," insisted the Shadow Brexit Secretary.

Jo Swinson, the deputy Liberal Democrat leader, on behalf of the People’s Vote, said: “Chequers is dead; it is official.

“The EU27 could not be clearer. Chequers is over whatever Theresa May might suggest."

The East Dunbartonshire MP said whether it was a version of the Canadian trade deal, a blind Brexit or a no-deal, Britain was heading for a “very bad deal” and the way out of the mess was a People’s Vote.