But despite May's appeal at an EU summit in Austria, European Council President Donald Tusk insisted that parts of her offer are still not satisfactory more than 18 months into the negotiations and must change to keep alive hopes of concluding a Brexit deal in coming weeks.
Mrs May promised new proposals to reassure Dublin that it would not end up with a "hard border" with the British province of Northern Ireland but warned that she too could live with a no-deal outcome - though many around the summit in Salzburg do not believe that that is a credible threat.
She added: "There is still a lot of work to do", and a no-deal scenario can't be ruled out. 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".
On the issue of keeping the Northern Irish border open and "frictionless", to avoid a visible border that would risk reigniting old tensions, May said the UK's proposal was "the only serious and credibly proposition on the table for achieving that objective".
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And although Tusk told reporters there were some "positive elements" in the UK's blueprint, there is still no agreement on the Irish border.
Donald Musk said that: "The suggested framework for economic cooperation will not work".
Mr Tusk said a Brussels summit on Oct 18 would be a "moment of truth" to overcome remaining big problems and leaders pencilled in the weekend of Nov 17-18 to formalise a final agreement.
Raab said the government is sticking to its proposal for a post-Brexit border between its province of Northern Ireland and European Union member Ireland, one of the major obstacles to an agreement.
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He said: "What the European Union is asking in and around Northern Ireland is actually impossible for the United Kingdom to accept".
Yet these hopes were largely dashed by a succession of negative verdicts on May's proposal for a free trade area with the European Union, which she drew up with her ministers at her Chequers country residence in July. So that's why we are cautiously optimistic that we will get a deal.
Mrs May was left visibly furious after French President Emmanuel Macron said Brexit had been sold to the British public by "liars" and that she needed to come up with "new propositions" if she wants to salvage a deal.
However, May warned it would not happen under her leadership, saying: "The UK will leave on March 29 next year".
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Irish Prime Minister Leo Varadkar complained that May had not presented any fresh proposals for the Irish border, despite promising them.
Speaking after her meeting with Mr Tusk, the PM said: "We both agree there can be no withdrawal agreement without a legally operative backstop". But she would not say when the proposals would be made public.
However, May signalled that Britain may be willing to compromise on another sticking point in the talks, on a fall-back plan to avoid frontier checks until it can be resolved through a wider trade deal.
Attention had been increasingly focused on the prospect of a special summit in November to finalise an agreement, however Mr Barnier said it should be clear before then whether a deal was actually possible.
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