May had met Tusk after the other European Union leaders had discussed Brexit in her absence over lunch, amid signs that they were attempting to harden their stance. But she would not say when the proposals would be made public.
But, asked about support for another referendum, an EU source said on Thursday: "The EU certainly does not have a position on that". "A good relationship at the end of this process depends on it". "Every day that is left we must use for talks", Tusk said.
May says her so-called Chequers plan will guarantee the frictionless movement of goods across borders, and ensure there is no return to a hard frontier between Ireland and Northern Ireland. European Union leaders in Salzburg repeated their view that the plans would undermine their cherished single market.
Mr Tusk said: "In October we expect maximum progress and results in the Brexit talks and then we will decide whether conditions are there to call an extra summit in November to finalise and formalise the deal".
"But we all agreed on this today, the proposals in their current state are not acceptable, especially on the economic side of it", he said.
"It must be clear that there are some issues where we are not ready to compromise".
"Europe isn't an a la carte menu", French President Emmanuel Macron said. "I say these words as a close friend of the United Kingdom and a true admirer of PM May". The EU has continued to reject this idea, however, and it may now be back to the drawing board for May's fractious government. However, she refused to deny the new plans would involve regulatory checks in the same way.
She has flatly rejected a European Commission backstop proposal for Northern Ireland to remain within the EU customs area after Brexit, arguing this would draw a border down the Irish Sea.
"The Irish question remains our priority too and for this, we need not only goodwill, which we feel, the atmosphere was better than two or three weeks ago, but the Irish question needs something more than good intentions", he noted.
"We've already compromised hugely with the Chequers proposals".
Brett Kavanaugh Accuser & Anita Hill Head To Emmys, In Name At Least
He continued, "Because this never happened, I had no idea who was making this accusation until she identified herself yesterday". Susan Collins, R-Maine, tweeted Monday that Kavanaugh and Ford "should both testify under oath before the Judiciary Committee".
She also hit back at whispers in Brussels of a referendum on the final Brexit deal, saying: "We all recognise that time is short but delaying or extending these negotiations is not an option".
For the British media, the message from Salzburg was clear.
The Brexit-supporting tabloid Sun branded bloc leaders "EU dirty rats", accusing "Euro mobsters" Tusk and Macron of "ambushing" May.
Reports suggest that European Union officials are keen not to appear too hostile to May, or box her into an even more entrenched position - aware that she faces a daunting task to placate many doubters within her ruling Conservative Party, and to persuade a reluctant parliament to back a deal.
"That deal of course would involve three elements, so withdrawal agreements providing for the United Kingdom to leave with a transition period, a protocol on Ireland and Northern Ireland making sure that under all circumstances there won't be a hard border on the island of Ireland, and also a joint political declaration outlining what that new relationship is going to look like".
"Theresa May's Brexit negotiating strategy has been a disaster", opposition Labour leader Jeremy Corbyn said.
However, May warned fellow leaders it would not happen, nor would Britain consider delaying Brexit to allow more time for negotiations. "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".
Scottish First Minister Nicola Sturgeon called May's statement "dreadful" and warned that May's party would pay a high political price if there was "no deal".
"The other camp would like to deliver evidence that to make that kind of decision is to be punished and that the British must suffer", he said.
Many are pushing for a new referendum that would let voters choose between accepting whatever deal she manages to negotiate with the bloc and staying in the EU.