The Salzburg summit had "triggered a crisis in government", said The Times, adding that ministers could force her to drop the so-called "Chequers" plan - which envisages a UK-EU free trade area just for goods through a common rulebook - "within days".
In a speech that was briefly delayed by a power outage in 10 Downing Street, May said, "I have always said these negotiations were going to be tough, and they are going to be tougher in the final stretch".
It warned the bloc against making "unacceptable" demands on the Irish border, repeating her mantra that no UK PM could ever accept Northern Ireland being under a foreign customs jurisdiction.
"I have treated the European Union with nothing but respect".
May said Friday that Britain would set out alternative plans dealing with the thorny issue of the Irish border, "that preserves the integrity of the UK".
"Neither side should demand the unacceptable of the other, we can not accept anything that threatens the integrity of our union as they can not accept anything that threatens the integrity of theirs". She's facing increasing calls to dump her vision of what the future trading relationship should be, and speculation is rife that some in her party will try to oust her. So we are at an impasse.
Mr Tusk, who has insisted Britain can not "cherry-pick" benefits of European Union membership to retain, took a pop at the prime minister by putting a photo of them choosing cakes at the summit on Instagram.
"We want to ensure predictability for our people and businesses so we want to somehow ease, mitigate the sometimes very tense and hard negotiations for Britain's exit from the European Union because we really want to break the impasse" between them, Morawiecki added.
"At this late stage in the negotiations it is not acceptable to simply reject the other side's proposal with no counter offer".
"Theresa May's Brexit negotiating strategy has been a disaster", opposition Labour leader Jeremy Corbyn said.
In response to May's statement, the Confederation of British Industry and other business bodies said they wanted to see constructive dialogue, not rhetoric.
"Everybody shared the view that while there are positive elements in the Chequers proposal, the suggested framework for economic cooperation will not work", Mr Tusk said. The U.K., she said, would be waiting for the E.U.to come back with reasonable proposals.
With a defiant tone, May concluded by saying: "We need serious engagement on resolving the two big problems in the negotiations and we stand ready". British officials had been hoping for warmer words from European leaders at the Austrian event - to bolster Mrs May as she prepares for her party's conference in a fortnight. When the Conservatives meet for their annual conference on September 30, they plan to push for May to ditch the Chequers plan, or face a challenge to her leadership.