May will attend a European Council summit in Brussels this week to discuss the plan for a Brexit extension.
French Foreign Minister Jean-Yves Le Drian said a delay could only be granted if May guaranteed that its objective "is to finalize the ratification of the deal already negotiated". "I agree. I am on your side".
But the deal has twice been rejected twice by hefty margins in Britain's Parliament, amid opposition from pro-Brexit and pro-EU lawmakers.
"Even if the hope for final success may seem frail, even illusory, and although Brexit fatigue is increasingly visible and justified, we can not give up seeking until the last moment a postivie solution", Mr Tusk said.
While May wants an extension until the end of June, EU leaders do not want the delay to go beyond May 23 if the United Kingdom is to avoid having to participate in forthcoming European Parliament elections.
Former attorney general Dominic Grieve, who is opposed to Brexit, told the Commons the sight of May rebuking Parliament for its role in the process was "the worst moment I have experienced since I came into the House of Commons".
"Although Brexit fatigue is increasingly visible and justified, we can not give up seeking until the very last moment a positive solution", he said.
Confirming a short extension will be requested, a Number 10 source said: "PM won't be asking for a long extension".
In briefings, diplomats from several European countries made it clear that they would do everything they could to avoid having Britain leave without the safety of a withdrawal agreement.
However, analysts suggest that far from increasing the prospect of a "no deal" exit, Tusk's ultimatum could strengthen May's hand - as well as that of campaigners demanding a second referendum which they hope will reverse Brexit.
'Motion after motion and amendment after amendment has been tabled without Parliament ever deciding what it wants.
She now looks set to try and bring her deal back for a third meaningful vote next week, despite a ruling by Speaker John Bercow that it must be changed before it can be voted on again.
Although people involved in the negotiations said that there was little appetite for ejecting Britain against its will, they were fearful that further surprises in London could lead to what one senior European Union diplomat called "catastrophe".
As a result, we will now not leave on time with a deal on 29 March.
We asked you the question already and you gave us your answer.
European Commission President Jean-Claude Juncker signalled on Wednesday that EU leaders may have to hold an emergency summit next week if there is a dramatic change of course at a minute to midnight. "And I think even God sometimes reaches a limit to his patience". In a letter to European Council President Donald Tusk, May asked for more time to get a Brexit deal passed.
Meanwhile, a SkyData poll found that 36 per cent of people blame Mrs May for the Brexit fiasco, 26 per cent point the finger at MPs, 7 per cent are irate at the European Union, while 24 per cent blame all of them equally. May ignored the warning.