The leaders of the other 27 European Union member states will discuss May's proposals without her over a working dinner on Thursday.
Britain and its European Union partners failed on Thursday to secure a breakthrough in Brexit talks, largely because of seemingly intractable divisions over the best way to avoid a hard border on the island of Ireland and how to deal with future trade.
After receiving what she described as a "frank" briefing on the Brexit talks from Tusk, May gave a defiant press conference in which she insisted her plan was "the only proposal on the table".
But, she added, "that backstop can not divide the United Kingdom into two customs territories, and we will be bringing forward our own proposals shortly".
Writing in Die Welt daily, May said both sides needed to show goodwill to avoid a disorderly United Kingdom exit from the bloc next March, which some fear would unleash major economic disruption.
"Without clear a precise solution to the Irish question, and for the whole context of our economic future relations, it will be hard even to imagine a positive process after October", Tusk said.
In a brief statement in Salzburg on Wednesday, Tusk said that while proposals hashed out by May and her Cabinet at her Chequers country retreat are a "positive evolution", the issues surrounding the Irish border and economic cooperation need to be "reworked and further negotiated".
Future UK Prime Ministers must not be allowed to unpickTheresa May's Brexit deal after she leaves No 10, Brussels has warned.
Tusk said he would call an additional summit in mid-November to seal any deal with Britain.
"On the economic partnership, there is no solution that will resolve the Northern Ireland border which is not based on the frictionless movement of goods".
We are clarifying which goods arriving in Northern Ireland from the rest of the United Kingdom would need to be checked and where, when and by whom these checks could be performed.
The pro-Brexit Daily Telegraph splashed a picture of French leader Emmanuel Macron, identified as one of May's most ruthless critics, on its front page, along with the sub-head "Disastrous negotiations threaten PM's leadership ahead of Tory conference".
According to one person familiar with the matter, the special summit will be called for November 17-18 - a weekend - but leaders chose to keep it conditional on more progress in an effort to add to the pressure on May to concede further ground.
Mrs May said: "I welcome the fact that Michel Barnier is recognising the need to find a new solution because the original proposal put forward by the European Commission was unacceptable to us". A special Brexit summit could still be set up in mid-November if things progress as hoped.
While some progress has been made in recent weeks on negotiations about the terms of the Brexit withdrawal agreement, negotiations over Britain's future relationship with the European Union remain stuck over May's plan to keep aligned with the European Union market in goods, but not in services.
Mrs May will be hoping for a diplomatic breakthrough and some harmony at the Mozarteum.
May says she can not accept having customs checks within the United Kingdom.
"But our European partners if they want to reach a deal, have to find a way of understanding that there are some things we can not accept".
Mrs May has just told a clammy, tiny room full of journalists: "I believe that there is willingness to do a deal but let nobody be in any doubt, that as I've always said, we are preparing for no deal".