But Tusk was clear - her customs proposals and those aimed at preventing a hard border on the island of Ireland would have to be reworked. Earlier, EU negotiator Michel Barnier signalled he was ready to address Britain's border concerns.
"The Brexit negotiations are entering their decisive phase. Various scenarios are still possible today but I'd like to stress that some of Prime Minister May's proposals from Chequers indicated positive evolution in the UK's approach," Tusk told a news conference.
"On other issues, such as the Irish question, or the framework for economic cooperation, the UK's proposals will need to be reworked and further negotiated."
A British government source said London welcomed the EU's commitment to finding a solution to the border issue, but could not accept any proposal that would effectively move the customs frontier into the Irish Sea, separating Northern Ireland from the rest of the United Kingdom.
With just over six months to go till Britain leaves the EU, time is pressing and both sides are keen to secure some kind of deal before the end of the year and ease concerns over a disorderly Brexit, which could push the economy into a downturn.
Writing in Germany's Die Welt, May said the two sides were "near to achieving the orderly withdrawal that is an essential basis for building a close future partnership".
"To come to a successful conclusion, just as the UK has evolved its position, the EU will need to do the same.
"With goodwill and determination on both sides we can avoid a disorderly exit and find new ways of working together."
'SOUND OF MUSIC'
At a summit dinner at the Felsenreitschule theatre -- known to film fans for a scene in the musical 'The Sound of Music' -- May will make another pitch for support for her Chequers plan, named after her country residence where a deal was hashed out with her ministers earlier this year.
Then she will be out of the room on Thursday afternoon when the other 27 leaders discuss her Brexit proposals.
Tusk said he would call an additional summit in mid-November to seal any deal with Britain.
While both sides have been making positive noises, how to keep an open border between Northern Ireland and the Irish republic remains one of the biggest obstacles.
"On Brexit, it's very difficult. We have nothing on Ireland," Jean Asselborn, Luxembourg's foreign minister, told Reuters.
Northern Ireland's Democratic Unionist Party, which backs May's minority government in Westminster, criticised Barnier's ideas for handling customs crossing the Irish frontier, saying they would still mean an unacceptable border between the province and the rest of the United Kingdom.
The government source said: "We have been very clear that what we cannot accept is Northern Ireland being separated off from the UK customs territory."
But EU officials are minded not to paint May into a corner, aware that she faces increasing opposition to her plans in her Conservative Party, and needs a victory of sorts to persuade a reluctant parliament to back a deal.
May has told lawmakers that they will vote either for a Chequers-based deal or to leave without an agreement. Mel Stride, a junior treasury minister, said pro-Brexit and pro-EU lawmakers could usher in a second referendum on EU membership if they reject her plan.
(Additional reporting by Alastair Macdonald, Francois Murphy Andreas Rinke, Joseph Nasr in Berlin Writing by Elizabeth Piper Editing by David Stamp) ((email@example.com; 07979746994; Reuters Messaging: firstname.lastname@example.org))