Britain's Prime Minister Theresa May addresses a press conference at the end of the EU Informal Summit of Heads of State or Government at the Mozarteum University in Salzburg, Austria, on September 20, 2018.
May promised new proposals to reassure Dublin that it would not get a "hard border" with the British province of Northern Ireland but warned she too could live with a no-deal outcome - though many round the summit table in picturesque Salzburg see that as more of a negotiating tactic than a credible threat.
"I want to be absolutely clear, this government will never accept a second referendum", she said.
But at a press conference in which she appeared visibly angry, Mrs May said the United Kingdom would "shortly" be coming forward with new proposals for the backstop at the border.
After the summit, European Union leaders said they would push for an agreement next month, but both sides have warned they are planning for a no-deal scenario.
But the European Union has reservations about the plan, and the two sides remain divided over how to ensure there is no hard border between the U.K.'s Northern Ireland and European Union member Ireland.
"Without clear a precise solution to the Irish question, and for the whole context of our economic future relations, it will be hard even to imagine a positive process after October", Tusk said.
In a brief statement in Salzburg on Wednesday, Tusk said that while proposals hashed out by May and her Cabinet at her Chequers country retreat are a "positive evolution", the issues surrounding the Irish border and economic cooperation need to be "reworked and further negotiated". "This is why we need more time, but we hope to be ready in October", he said, dubbing the planned October 18 European Union summit the "moment of truth".
After a dinner of Wiener schnitzel in Salzburg, EU leaders said they would push for a Brexit deal next month but rejected May's proposal.
However, at the gathering in Austria, Mrs May insisted there would be no delay to the UK's March 2019 departure, no second referendum and therefore the onus was on the continent's leaders to find a solution if they wanted to avoid a no-deal Brexit.
European Union leaders say that is "cherry-picking" on the part of Britain, which would be able to go its own way in services and on other issues such as the freedom of movement of labour.
Leave-backing Transport Secretary Chris Grayling insisted Britain is "preparing hard" for a no-deal Brexit, although he predicted a "sensible agreement" will be reached with Brussels.
"We need to compromise on both sides", Tusk noted.
The EU has insisted that the single market can not be cherry-picked like that.
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.
Tusk added he remains "convinced that a compromise, good for all, is still possible" in Brexit negotiations and that he is "a close friend of the United Kingdom and a true admirer of PM May". This would keep Northern Ireland under EU economic oversight if London and Brussels can not agree a trade pact to keep UK-EU borders open after a transition period ends in 2020 - an idea that Mrs May and a small party in the province that props up her minority government oppose.
Theresa May has been warned Britain is "staring down the barrel of no deal" after she issued a defiant Brexit challenge to the EU.
Summit meeting: (left-right) Leo Varadkar, Commission President Jean-Claude Juncker and and Estonian Prime Minister Juri Ratas.
Before setting off, her Downing Street office suggested that after she put forward new Brexit proposals this summer, it was now for Brussels to "evolve" its position.
May said that leaving with no deal remains an option.