But European Council president Donald Tusk stressed that economic aspects of Mrs May's Chequers proposals "will not work", while former Brexit secretary David Davis disclosed a "rock-solid core" of up to 40 Tory colleagues could vote against the Government's favoured approach.
Tusk said Thursday after a two-day summit partly centering on the British divorce talks with the EU that some of the suggestions from Prime Minister Theresa May would undermine the union of the 27 remaining members.
The backstop would keep Northern Ireland under EU economic oversight if London and Brussels can not agree a trade pact to keep UK-EU borders open, an idea that May and a small party in the province that props up her minority government oppose.
"If the political will is there on the other side I am confident we will reach a deal which is in the EU's interests as well as the UK's", she said.
With barely six months until Britain leaves the bloc, at the risk of serious disruption if there is no deal to tie up legal loose ends, there is pressure on both sides: "You can hear very clearly the clock ticking in the room", said the second diplomat.
The UK Prime Minister told reporters she and Tusk had a "frank bilateral" with Tusk after the summit-closing press 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", Tusk said.
Tusk said a Brussels summit on October 18 would be a "moment of truth" to overcome remaining big problems and leaders pencilled in the weekend of November 17-18 to formalise a final agreement.
It comes as supporters of a second referendum publish a blueprint setting out how MPs could force the Prime Minister to accept what they call a People's Vote on her deal, with the option of remaining in the EU.
UK Leader Warns Rebels: It's my Brexit Agreement or no Deal
The EU had been suggesting a hard border could be avoided if Northern Ireland continued to function as if it was within the EU but, in an article in German newspaper Die Welt, May said such an arrangement would effectively divide the UK.
But Tusk said that was now contingent on May reworking her Chequers plan by Ocotber.
European Union summit chairman Donald Tusk repeated criticism of her proposals for future customs arrangements and for the border between the British province of Northern Ireland and European Union member Ireland, saying these needed to be reworked.
"What I'm not going to do is to say that I would refuse to entertain any further proposals that the European Union comes up with but they've got to be respecting the equities that we've set out", he said.
"Europe isn't an a la carte menu", French President Emmanuel Macron said.
"Unfortunately we can not at this stage exclude a no-deal - it depends on both sides of negotiations".
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 Financial Times said Barnier's latest comments suggest he wants to reduce tension over the issue ahead of the summit in Salzburg on Wednesday. We must do everything to avoid a hard Brexit.
She insisted it is the only way to provide the "frictionless" movement of goods on the frontier between British-ruled Northern Ireland and the Irish republic, which both sides have pledged to keep open.More news: Amazon aims for music lovers with new lineup of Echo devices