Mrs May promised to draw up alternative proposals to unblock the Irish border issue ahead of a crunch summit on October 18 - described as the "moment of truth" by Mr Tusk. "A piece of cake, perhaps?"
The negative headlines indicate the extent of the divergence in perceptions between London and the capitals of the EU's other 27 members on the future of Brexit.
Energy-rich Norway opted against joining the EU in 1994, choosing instead to enter the European Economic Area (EEA) which gives it all the benefits of the single market. But it would abandon the bloc's right to the free movement of people and see Britain adopt its own migration policy.
1 When is the Brexit deadline?
Liberal Democrat leader Sir Vince Cable said the Chequers plan was "dead as a dodo, killed in London by Tory fundamentalists", while Green MP Caroline Lucas said Mrs May's response "pathetic, painful and petulant".
British Prime Minister Theresa May's aides have begun contingency planning for a snap election in November to save both Brexit and her job, the Sunday Times reported. The EU's view of the Chequers plan was bluntly and clearly expressed in Salzburg yesterday, but should not have come as a surprise to the UK Government, given that it was not a new position.
But he added: "Even in the case of no deal I can not imagine that they would not reach some kind of pragmatic adjustments to smooth the exit and once that start happening then a pragmatic deal would be reached after all".
2 What is the transition period?
The rocky European Union summit dashed British hopes of a breakthrough in stalled divorce talks, with just six months to go until Britain leaves the bloc on March 29. If there is no divorce agreement, there will be no transition.
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"With just four weeks until a hoped-for final deal and with the issue of the Irish border still deadlocked, the chances of a no-deal Brexit in March have risen further over the past 48 hours", said James Stewart, KPMG's head of Brexit analysis.
Mrs May called on the European Union to provide counter proposals.
There are fears that frontier checks would disrupt trade.
But while Brexiteer MPs lauded her for standing up to the European Union there were also fresh calls to abandon her Chequers plan - which many Tories believe would keep Britain too close to the European Union - for a simple Canada-style free trade deal.
She said this would lead to the separation of Northern Ireland from the rest of the United Kingdom with an effective border down the Irish Sea.
Although Mrs May has said she would be willing to make a compromise on the Irish border, there are doubts it will go far enough to win support from the bloc.
Thus he commented on the image of the Prime Minister of nice Britain Theresa might and EU President Donald Tusk, that Tusk has revealed in his instagram.
The pound plummeted following Mrs May's speech, giving up four days of gains on the U.S. dollar to shed 1.5% and close at 1.30 in London.
"What Theresa May is saying is "don't mistake British politeness for weakness", he told BBC Radio 4's Today programme. One person in her inner circle told cabinet members she'll probably stand down next summer in a move created to stop them from resigning now in protest at her leadership.