He delivered his verdict at the end of a summit in Salzburg where the Prime Minister appealed directly to other European Union leaders for further compromise in the search for a deal over the terms of Britain's exit.
In a press conference just minutes after Tusk publicly shot down her plan, a visibly angry May called on the European Union to put a counter-offer on the table that would keep the border between Ireland and Northern Ireland "frictionless".
"My experience so far within the context of the EU Council is that, irrespective of one's political allegiances, there is a lot of respect [for UK Prime Minister Theresa May]", Muscat said in an interview with BBC Radio 4. "The Chequers plan can not be take it or leave it", Macron said.
Barnier hopes the Salzburg summit can set the process on to a countdown to a deal in two months - diplomats said leaders agreed to hold their diaries for a possible final Brexit summit in Brussels on the weekend of November 17-18.
Carmakers are anxious that port and motorway holdups could slow the movement of components and finished models, crippling output and adding costs, if Britain fails to reach agreement with the European Union over its departure from the bloc on March 29.
It's time for a people's vote on Brexit, with the option to stay and lead reform from within the EU. But May said a divorce deal is within grasp if both sides show "good will and determination".
"We need to get into the nuts and bolts of how this backstop can work and how the concerns that people have can be accommodated", Coveney told the Financial Times on Thursday.
May has been urging her European counterparts to accept her controversial Chequers plan, which has divided the Conservative party and the United Kingdom in general after it was thrashed out back in July.
The UK industry - already feeling the effects of a crackdown on diesel-powered cars in its largest European Union markets amid jittery demand - fears Brexit could result in lost access to vital European Union workers and massive costs from tariffs and additional red tape including supply chain disruption.
The response of Tusk left May on the defensive, giving her an hour to prepare for a press conference in which she appeared visibly nervous, and forced to repeat that her proposals could still form the basis of negotiations at the EU's October summit and a likely emergency summit in November.
If MPs rejected the Brexit deal, or no final deal was reached by January 21 next year, the government must table a statement on how it wants to proceed, and Parliament could use the vote that would take place on this as an opportunity to call for a referendum.
Assuring the media that the EU was also prepared, European Commission president Jean-Claude Juncker said: "Don't worry, be happy".
However not everyone is happy with the plan and some of her most senior ministers resigned because they didn't agree with it.
Keir Starmer MP, the Shadow Brexit Secretary, 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".
"We can not afford a "no deal" Brexit".
"I think Chequers now has no supporters at all".
"The government will not accept a second referendum because there has been a vote of the people".