Britain is due to leave the European Union on March 29, yet little is clear: there is, so far, no divorce deal, rivals to May are circling and some rebels have vowed to vote against a possible Brexit deal.
Tusk said the "moment of truth" on the negotiations was less than a month away, at the October meeting of the European Council.
A Downing Street source quoted by Reuters said the prime minister would use the Salzburg summit to insist that an agreement must respect her country's integrity, "which the Commission's proposal does not".
"We need to compromise on both sides", said Tusk, highlighting the importance of the October 18-19 summit in Brussels.
Speaking to the media in Salzburg on Thursday, French President Emmanuel Macron said the staunch Brexit backers had deceived the British people, promising them a windfall of funds and a painless exit.
Chancellor Angela Merkel agreed, saying the tone of discussions at the Salzburg summit showed the "atmosphere" of Brexit negotiations were improving.
He said: "It was a good and courageous step by the Prime Minister".
"It's either a so-called hard Brexit on WTO, which I'm perfectly comfortable with - but I am not sure the House of Commons is - and then we are into whether there will be an extension of Article 50 which would be a huge betrayal of the electorate". "We are hiring extra staff and officials, bringing in IT system", he said.
"We're ready for that eventuality, should it occur". The EU has insisted that the single market can not be cherry-picked like that.
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In a series of tweets, RACQ Rescue confirmed it had sent the rescue helicopter to Cid Harbour to retrieve the victim. The incident occurred within 24 hours of an attack on a Tasmanian women in the same area.
In another blow to May's hopes of winning over European Union leaders, French president Emmanuel Macron told reporters that Brexit was a project which had been pursued by "liars".
"Everybody shared the view that while there are positive elements in the Chequers proposal for economic cooperation, the suggested framework will not work - not least because it risks undermining the single market", Tusk told reporters.
Both sides have pledged to avoid a "hard" frontier in Ireland, amid fears it could disrupt trade and upset the fragile peace on the island.
The British government says the EU's fallback plan would effectively keep Northern Ireland in a customs union with the EU after the rest of the United Kingdom leaves - creating a border down the Irish Sea.
Transport Secretary Mr Grayling told BBC Newsnight they had put forward the only proposal that meets the UK's red lines and also provides the "right solution" for the Irish border. But the latter must be dealt with as part and parcel of the withdrawal accord.
However hard Brexiter David Davis, the former Brexit secretary who resigned in protest over the Chequers plan, predicted on Thursday morning the European Union would push Britain into a deal that would "likely look an bad lot like [EU] membership".
Macron acknowledged that Brexit was the choice of the British people but added that it was "pushed by those who predicted easy solutions".
He said Brexit "demonstrated that those who explain that one can get by easily without Europe - that it will all go well, that it is easy and brings in a lot of money - are liars".
"They build up, there's tough language and actually a deal is done at the last".
Tusk held open the prospect of an "extraordinary summit" in November to formalise the deal.