In London, Tusk met May for their first head-to-head meeting since Britain voted to leave the bloc in a referendum on June 23 which led to the resignation of her predecessor David Cameron. Tusk and May are keen to discuss what steps might be taken over the next few months, officials said.
May has said Britain will not trigger Article 50 of the EU's Lisbon Treaty to start the exit procedure this year to give her government time to come up with a negotiating stance for the complicated talks that will shape the country's future standing.
"Our goal (is) to establish closest possible EU-UK relations. Ball in UK court to start negotiations. In everybody's best interest to start ASAP (as soon as possible)," Tusk said on Twitter.
As head of the European Council, which groups heads of EU states and governments, Tusk leads the body that defines the bloc's political direction and priorities.
EU officials are keen to move quickly on the talks, fearing uncertainty over future relations is hurting investment.
But some data suggests that Britain's economy, while slowing sharply, has recovered from the initial impact of the vote. Reports published on Thursday showed firms increasing the number of permanent staff and house prices rising, albeit from low levels.
Lawmakers who had lobbied for Britain to leave the EU in the run up to the referendum have taken the economic data as proof that the 'remain' campaign had tried to frighten voters into staying by forecasting economic difficulties.
One, Liam Fox, who is now trade minister, said Britain was pressing on with plans to reach agreements with some of the world's largest and fastest-growing economies. He told parliament the government had set up a working group with India.
On Wednesday, May's spokeswoman said the meeting with Tusk would not only cover Britain's exit, or "Brexit", but would also discuss issues on the agenda for the October meeting of EU leaders, suggesting that Britain still plans to play a role.
May has said she will not show her hand before starting the Brexit talks, giving few details of what her government wants when it leaves the EU.
She says reducing immigration into Britain is crucial after millions of Britons expressed their frustration in the vote over what they say is the stress on schools, hospitals and housing from high numbers of people settling in the country.
But May, a former interior minister who was in charge of the ruling Conservative Party's immigration policy, also says she wants the best trade deal for Britain, refusing to say whether the country will remain in the EU's lucrative single market.
Banner and thumbnail credit: Reuters, Francois Lenoir