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.
Adding: "It's even more true since they left the day after so as not to have to deal with it". But EU leaders clearly were not feeling nostalgic.
But May said that after she presented her so-called Chequers plan on post-Brexit trade ties this summer, Brussels must also give ground.
But Mr Tusk dashed any hopes of a reprieve, saying after a working lunch to discuss the UK's plans: "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 is undermining the single market".
On the issue of keeping the Northern Irish border open and "frictionless", to avoid a visible border that would risk reigniting old tensions, May said the UK's proposal was "the only serious and credibly proposition on the table for achieving that objective".
The pound had earlier risen to a nine-week high, helped by strong United Kingdom inflation data and as investors prepared for the European Union summit where diplomats will try to pave the way for an agreement on Brexit. They worry about any diminution in a Brexit deal of the "four freedoms" that are the basis for the European Union - including the single market for seamless trade and freedom of movement and labor.
President Emmanuel Macron of France was scathing.
And, possibly for that domestic audience, she told the European Union leaders that although time was short, "delaying or extending these negotiations is not an option" and rule out the option of a second referendum on Britain's European Union membership.
"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.
Macron called May's proposals "not acceptable in their current state".
Firmino strikes late for Liverpool
When PSG equalised in the 83rd minute, I was thinking "well, this is a draw now" but Liverpool still pushed forward for a victor .
The UK could gain special access to the single market but might have to, in return, compromise on immigration agreements.
But Tusk said that was now contingent on May reworking her Chequers plan by Ocotber. He said he was seeking a deal with Britain that protects the interests of "the other 425 million people" who remain in the European Union.
An EU official said: "Things did not happen as we expected".
Mr Coveney said: "The consequences of a no-deal Brexit are so damaging and so severe, in particular for Britain, but also having a negative impact on Ireland".
He added: "I've not seen these two camps at the table".
"If you want to make a deal both sides will need to try to find a compromise", he said.
He said he was working on an "improved" plan to avoid the return of a hard border while respecting the territorial integrity of the United Kingdom, in an attempt to break the deadlock. The Polish former leader said the atmosphere in talks between May and the other 27 leaders had been better than before but that stark differences remained on trade and on the question of the Irish border.
"But she will get up tomorrow morning and she will keep going, and I think the reserves that she has - she never talks about being type 1 diabetic, she never talks about now being over 60 or any of that sort of stuff - but the resilience she is showing under the most extreme pressure, we haven't seen in politics since the financial crash of 2008 and before then for a couple of decades before that".
Ruth Davidson has challenged those in the Tory party who want to oust Theresa May from Number 10 to "put their sharp elbows away" and back her as she battles to deliver an Brexit deal for the UK. But leaders were also told to block out a weekend in mid-November for an emergency meeting, if required.