And although Tusk told reporters there were some "positive elements" in the UK's blueprint, there is still no agreement on the Irish border.
The backstop would keep Northern Ireland under EU economic oversight if London and Brussels can not agree a trade pact to keep UK-EU borders open, an idea that May and a small party in the province that props up her minority government oppose.
The EU insists on its own "operational and legally-binding Irish backstop" - what it describes as an insurance policy to prevent the return of physical infrastructure on the border in the event no other solution can be found. However, they hope their negotiator, Mr Michel Barnier, can secure her agreement next month to what will be new European Union proposals that will be fundamentally unchanged but may be politically more palatable, especially on Northern Ireland.
But the United Kingdom insisted this was unacceptable as it would split Northern Ireland off from the rest of the UK.
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's "Chequers plan" - named for the prime minister's country retreat where it was hammered out in July - aims to keep the U.K.in the European Union single market for goods but not services, in order to ensure free trade with the bloc and an open border between the U.K.'s Northern Ireland and European Union member Ireland. Jaguar cited uncertainty over Brexit as one reason for putting 1,000 workers on a three-day work week until Christmas.
It feels that the search for something else has been in vain.
"They're the things that I can't compromise on".
But to kick out publicly as they did in Salzburg certainly runs the risk of pushing Mrs May too far.
She promised that the United Kingdom would be putting forward its own proposal for the Irish border soon.
Earlier, the leaders of the Czech Republic and Malta had suggested a second referendum should be held to give the British people the chance to rethink Brexit. "And actually if they stick with that position, there will be no deal", he said.
Britain is set to unveil new proposals on the future status of the Irish border, in a bid to break deadlock in Brexit negotiations.
But even countries that are basically sympathetic to Britain, like the Netherlands, Denmark and Sweden, have been firm supporters of the guidelines given to EU's Brexit negotiator, Michel Barnier.
"They build up, there's tough language and actually a deal is done at the last".
But he added negotiations were not yet at an "end game" and said Chequers was "broadly" a package that could get through Parliament.
May's former Brexit minister David Davis has said up to 40 lawmakers from the Conservative Party will vote against her Brexit plans.
He told HuffPost UK that, with Labour also planning to vote against Mrs May's plan, the "critical size of the voting bloc is quite small, it's basically a dozen people". Mr Gove, who was a major face of the Leave campaign, urged Tory colleagues to support Mrs May's Chequers plan because a future PM could "choose to alter" Britain's post-Brexit ties.
A "no deal" Brexit, bringing uncertainty and insecurity for expatriates, has never looked more likely, however the EU27 leaders have agreed to push back the deadline for a withdrawal deal to be agreed to a one-off summit in mid-November since it is now clear the previous deadline of October 18 will be missed.
"If the political will is there on the other side I am confident we will reach a deal which is in the EU's interests as well as the UK's", she said.