Ms Sturgeon warned that it would be reckless for Britain to leave the European Union without a deal in March and has written to opposition leaders urging them to back a delay if no agreement has been reached.
He added that "tough language" was to be expected.
"The UK will leave on March 29 next year", she said, according to officials.
"But it is a mistake because we don't have the political space or inclination to change."Mrs May was defiant in a tetchy press conference, insisting that her model of a UK-EU free-trade area was the only credible proposal on the table".
Following two days in Salzburg, the British prime minister appeared to stick to her guns on the blueprint which is centred on a "common rulebook" approach, arguing that it was the only proposal that would avoid a hard Border in Ireland.
The EU has offered to effectively keep Northern Ireland in its customs union.
Mrs May said Chequers was "the only credible and negotiable plan on the table that delivers no hard border in Northern Ireland and also delivers on the vote of the British people".
Both sides had been aiming for an October EU summit as the deadline to reach an agreement, to allow time for the deal to be ratified by British and European parliaments before Brexit in March.
That has helped sterling rally more than 4pc from 2018 lows hit in August when fears that Britain would crash out of the European Union without a trade deal spooked investors.
It feels that the search for something else has been in vain.
And, to make matters worse, May was forced to stand outside the meeting after the Prime Ministers, Presidents and Chancellors returned to discuss her plan.
The Dutch prime minister, Mark Rutte, one of the European leaders more sympathetic to Britain, described the Chequers proposal as "helpful but not sufficient". Pro-Brexit Conservatives, including former Foreign Secretary Boris Johnson, hate the Chequers plan, saying it would keep Britain tethered to the bloc, unable to strike new trade deals around the world.
"On the economic partnership, there is no solution that will resolve the Northern Ireland border which is not based on the frictionless movement of goods", she said.
"Meanwhile, a "blind" Brexit will simply kick all of the hard decisions into the long grass -but with the United Kingdom already out of the EU".
However David Davis, who quit as Brexit secretary after the Chequers plan was agreed by the cabinet, disagreed.
Alison McGovern, an MP with the People's Vote campaign also chimed in.
"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.
"Tusk said a Brussels summit on October 18 would be a "moment of truth" to overcome remaining big problems and leaders penciled in the weekend of November 17-18 to formalize a final agreement".
"The Brexit teaches us something - and I completely respect British sovereignty when I say that - it showed that those who say that we can easily live without Europe, that everything is going to be alright, and that it's going to bring in a lot of money are liars", he said.