Tusk issued the ultimatum after crunch talks with May on the sidelines of a Brussels summit, which the embattled British premier described as taking place in a "very positive atmosphere".
The "absolute" deadline gives May until a meeting with European Commission chief Jean-Claude Juncker on December 4 to make sufficient progress on the key Brexit divorce issues.
Failing that, the EU leaders will refuse at their next summit on December 14-15 to unlock the next phase of the Brexit negotiations, which are supposed to cover a future trade deal and a transitional arrangement after Britain leaves the EU in March 2019.
"Sufficient progress in Brexit talks at December EUCO (summit) is possible. But still a huge challenge," Tusk, a former Polish prime minister, said on Twitter after his hourlong meeting with May.
"We need to see progress from UK within 10 days on all issues, including on Ireland."
An EU source told AFP the talks were "long and honest", with Tusk setting December 4 as an "absolute deadline for the UK to make additional efforts" in time for EU Brexit negotiator Michel Barnier to formally declare enough progress.
"PM May agreed to this timeframe," the source told AFP on condition of anonymity.
The EU insists that Britain must tie up three thorny divorce issues -- a multibillion-euro exit bill, the Irish border and the rights of EU nationals living in Britain -- before there can be any talks on future relations.
"Particular attention was on how to ensure the support of Ireland to move to the second stage," said the source, adding that it was "still unclear" how Britain would meet Dublin's demand to avoid any return of border restrictions with British-ruled Northern Ireland.
- 'Positive atmosphere' -
EU leaders had been hopeful that May would bring new proposals on Ireland and on the exit bill, after senior British ministers agreed earlier this week to double the offer to a reported 40 billion euros ($48 billion).
May struck a positive note despite acknowledging there were "still issues" after her meeting with Tusk on the sidelines of a summit with six former Soviet countries.
"There's been a positive atmosphere in the talks and a genuine feeling we want to move forward together," she said.
Talks did not start for almost a year after the June 2016 Brexit referendum and since then have been virtually deadlocked, first on the exit bill, and then on Ireland as well.
In a bid to make progress, May also met German Chancellor Angela Merkel and the leaders of Belgium, Denmark and Lithuania to press her case that there can be no final divorce agreement without a deal on trade.
Her hastily arranged talks with Merkel, whose status as Europe's most powerful leader has been undermined by the collapse of German coalition talks, were "very constructive", May's office said.
Belgian Prime Minister Charles Michel said on Twitter after meeting with May that "clarity on our future relations is in everyone's interest," while Danish Prime Minister Lars Lokke Rasmussen said "I have decided to be optimistic about this."
- No 'hard border' for Ireland -
The Irish government, however, has pushed the Northern Ireland border issue to the top of the agenda, insisting that Britain must provide solutions now to prevent a return of checkpoints on a frontier long plagued by sectarian tensions.
"If progress isn't made in terms of more clarity and more credibility... in a way that prevents a hard border on the island of Ireland, well then we cannot move on to phase two," Irish Foreign Minister Simon Coveney said.
But Coveney acknowledged that a political crisis in Ireland over the deputy prime minister's handling of a police whistleblower case had left the government in a "precarious position".
Failure to reach a deal at the December summit would leave little time for trade talks, which the EU wants to wrap up by October to allow time for a deal to be ratified by national parliaments ahead of Brexit Day on March 29, 2019.
May said her presence at the so-called Eastern Partnership Summit with six former Soviet states showed that Britain was "unconditionally committed to maintaining Europe's security" despite Brexit.
She also took aim at Moscow, saying Europe must be "open-eyed to the actions of hostile states like Russia."
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