The UK has yet to come forward with a solution to replace the backstop, says the European Commission.
Downing Street said a meeting between Boris Johnson and Commission president Jean-Claude Juncker was “constructive”.
But in a statement after the working lunch, the Commission said no proposals had been put forward to replace the controversial Brexit policy.
Mr Johnson has called it “undemocratic” and said it needs to be removed from any deal he would do with the EU.
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The bloc has said it is willing to look at alternatives, but that an insurance policy like the backstop must be in place.
The backstop is a part of the existing withdrawal agreement to prevent a hard border returning to Ireland.
Both Mr Johnson and Mr Juncker – who met for the first time since the PM took office in July – agreed the discussions between the UK and EU “needed to intensify” and meetings “would soon take place on a daily basis”.
But regardless of the outcome, No 10 said the PM “would not request an extension and would take the UK out of the EU on 31 October”.
The pair were joined by the EU’s chief Brexit negotiator Michel Barnier and Brexit Secretary Steve Barclay at the meeting in Luxembourg.
The statement from the European Commission said the purpose of the meeting was to “take stock” of ongoing talks, and look to the “next steps”.
But it added: “President Juncker recalled that it is the UK’s responsibility to come forward with legally operational solutions that are compatible with the withdrawal agreement.
“President Juncker underlined the Commission’s continued willingness and openness to examine whether such proposals meet the objectives of the backstop.
“Such proposals have not yet been made.”
The statement said the Commission would “remain available to work 24/7” and said the upcoming European Council summit in October will be “an important milestone in the process”.
It concluded: “The EU27 remain united.”
The No 10 statement said Mr Johnson confirmed his commitment to the Good Friday Agreement – the peace deal brokered in Northern Ireland – and still had a “determination to reach a deal with the backstop removed, that UK parliamentarians could support”.
‘Whatever the outcome’
The BBC’s reporter in Brussels, Adam Fleming, says the UK’s proposals so far are seen as “vague and insufficient” by EU officials.
“There is still scepticism in Brussels that the UK is serious about getting an agreement,” he writes.
Last week MPs passed a law that would force the prime minister to ask the EU for an extension to the 31 October deadline if a deal was not agreed by 19 October.
But the prime minister’s official spokesman said: “The position of the PM is that we comply with the law, but that we are leaving on 31 October whatever the outcome.”
They also confirmed that the current date set for a transition period – the time for the UK and EU to negotiate their future relationship after officially leaving – of December 2020 would not be extended.
Over the weekend Mr Johnson told a newspaper that the UK would break out of its “manacles” like cartoon character The Incredible Hulk – with or without a deal.
And a Downing Street source said the PM would make clear to Mr Juncker that “he would not countenance any more delays” beyond 31 October.
The BBC’s political editor Laura Kuenssberg said the issue of whether the UK had the legal right to leave on 31 October – come what may – could end up in court.
The prime minister’s optimism about getting a revised Brexit deal is not shared on the EU side.
Yes, the talks with the UK are becoming more focused and homing in on specific ideas, such as how European rules on plant and animal health could apply in Northern Ireland after Brexit.
But the UK’s proposals are seen as vague, insufficient and – according to one official – “like last night’s mac and cheese with fresh parsley scattered on top”.
And there’s still scepticism in Brussels that the UK is serious about getting an agreement.
There are also conflicting views about the role of Jean-Claude Juncker, the soon-to-be former president of the European Commission.
In the UK, there’s a belief that he wants a deal to crown his final months in office but European diplomats question whether he would want one of his final acts to be forcing Ireland into a major climb-down over the backstop.
Reports have suggested Mr Johnson is considering a plan to keep Northern Ireland more closely aligned to the EU after Brexit, as an alternative to the current Irish backstop.
The backstop is the controversial policy in the existing withdrawal agreement, rejected three times by MPs, which would require the UK to follow the EU’s customs rules to ensure there are no physical checks on the border between Northern Ireland and the Republic.
The Democratic Unionist Party – which supports the Conservatives in Parliament – has rejected any plan that would see Northern Ireland treated differently to the rest of the UK.
The PM’s spokesman would not give details, but said the government had “put forward workable solutions in a number of areas”.
Writing in Monday’s Daily Telegraph, Mr Johnson said he believed he could strike a deal with the EU within weeks and was working “flat out to achieve one”.
“If we can make enough progress in the next few days, I intend to go to that crucial summit… and finalise an agreement that will protect the interests of business and citizens on both sides of the channel, and on both sides of the border in Ireland,” he wrote.
Many MPs have also questioned how serious the government is about getting a deal such as former justice secretary David Gauke who said “detailed proposals” had yet to be put forward.
“It still remains the case the UK government has not produced detailed proposals as to how it wants to replace the Irish backstop,” he told Radio 4’s Today.
Foreign Secretary Dominic Raab said the PM would stress he wanted a deal, but there had to be “some finality” to it.
He said claims from the EU side that the UK was dragging its feet were part of the “tactical posturing that goes on in any negotiation”.
He told Today that the UK had been clear the “anti-democratic backstop” had to be removed from the withdrawal current agreement, and the outline of future trading relationship set out in the political declaration had to be much more ambitious.
“The EU knows our position. Lots of the detail has been talked through at technical and political level,” he said. “The framework is very clear.
“But of course the nature of these negotiations is that there will be a tendency to rubbish things we put forward in order to exact further demands. We are not going to get involved in that.”