In the latest frustrated comment from the EU as Brexit dealings crumble, European Commission boss Jean-Claude Juncker said his job in Brussels was "hell".
The European Council president was speaking alongside Irish Taoiseach Leo Varadkar following talks in Brussels in which they discussed preparations for what Mr Tusk described as the "fiasco" of a no-deal Brexit on March 29.
"The EU 27 is not making any new offer", Tusk said, adding that the legally binding withdrawal agreement sealing the divorce, which May negotiated and backed vociferously, can not be renegotiated.
After their statements, Mr Varadkar was heard telling Mr Tusk "they will give you awful trouble in the British press for that" - but the European Council chief just nodded and laughed.
At the end of the press conference as Tusk and Varadkar shook hands, microphones picked up the Irish premier telling him: "They will give you awful trouble in the British press for this".
Theresa May is due to meet with European Commission President Jean-Claude Juncker Thursday, where according to United Kingdom media reports she is expected to make a last-minute plea for changes to the Brexit deal she negotiated with the bloc, but which was rejected by Britain's Parliament.
"We will not gamble with peace or put a sell-by date on reconciliation".
The Swiss banking group is the latest financial institution to disclose a material transfer of its business away from London ahead of Britain's departure from the European Union on March 29.
While Mr Varadkar met the EU's leaders, Mrs May held meetings with all five of the main Northern Irish parties at Stormont House in Belfast.
Tusk said the Irish border issue and the need to preserve the peace process remained the EU's "top priority". She hopes these changes will help her get it through the UK Parliament.
And he demanded that Prime Minister Theresa May bring "realistic suggestions" for how to end the impasse and prevent a no-deal scenario. "This is why we insist on the backstop", he added.
No matter, Tusk and Varadkar both laughed.
The Brexit chief for Northern Ireland's Democratic Unionist Party, Sammy Wilson, branded Tusk a "devilish, trident wielding, euro maniac".
He said "from rapidly growing creative firms trading with South Korea to specialist machinery firms trading with Mexico, these EU Free Trade Agreements have supported firms of all sizes to grow and could be lost overnight".
Labour Party foreign policy spokeswoman Emily Thornberry said such a postponement would allow "time to see whether the negotiations succeed or, if they do not, to pursue a different plan".
United Kingdom cabinet ministers have secretly held talks on plans to delay Brexit by eight weeks, the Telegraph newspaper reported late on Tuesday.