Speaking to MEPs, the European Council president suggested Britain could turn its back on the EU referendum should it want to.
Tusk’s comments come after the EU’s chief Brexit negotiator indicated that agreement on a free trade deal with Britain after it leaves the European Union will take years to complete.
Michel Barnier said the discussions would be very different from the first phase of the negotiations on the terms of Britain’s withdrawal and would entail “risks”.
Last month, Mrs May denied that she has let down Brexit voters, following a speech in Florence in which she called for a two-year transition period following Britain’s official exit from the EU.
In her 35-minute speech she said that Britain would honour its commitments to the EU budget during this time, and challenged accusations that Britain has failed to secure a single concession from the EU during the negotiations.
Since then she had told MPs there could be no transition period after Britain leaves the EU unless there was agreement on a trade deal.
In a Commons statement, the Prime Minister said she remained confident that a deal was possible following the latest EU summit last week in Brussels.
However, her remarks caused consternation among opposition MPs who warned the UK was heading for a “cliff-edge” break with the EU which businesses have been desperate to avoid.
On Monday, the head of Britain’s largest business lobby group said UK businesses feel a “growing sense of urgency” for a Brexit transition deal.
“This is real, this is urgent and a transition agreement by the end of the year would help enormously to keep investment and jobs in the country,” Carolyn Fairbairn, director general of the Confederation of British Industry (CBI) said in comments to the BBC.
Tusk stressed the importance in the unity of the 27 members who will remain in the EU after the UK’s departure.
“Ahead of us is still the toughest stress test. If we fail it, the negotiations will end in our defeat,” he said.
“We must keep our unity regardless of the direction of the talks. The EU will be able to rise to every scenario as long as we are not divided.
“It is in fact up to London how this will end: With a good deal, no deal or no Brexit. But in each of these scenarios we will protect our common interest only by being together.”