Luxembourg's PM has attacked Boris Johnson's approach to Brexit, calling the situation a "nightmare".
Xavier Bettel said the British government had failed to put forward any serious proposals for a new deal.
But Mr Johnson, who pulled out of a joint press conference with Mr Bettel because of noisy protesters, said there was still a good chance of a deal.
A government source said the gap the UK and Brussels needed to bridge to achieve a deal "remains quite large".
Mr Johnson was visiting Luxembourg to hold talks with the European Commission President Jean-Claude Juncker and the EU's chief Brexit negotiator, Michel Barnier, as well as Mr Bettel.
After the working lunch with Mr Juncker and Mr Barnier, Mr Johnson said he had been encouraged by the EU's willingness to engage with the UK in their shared desire to avoid a no-deal exit - but there had not been a "total breakthrough".
However, the European Commission said the PM had yet to present concrete proposals for it to consider and insisted any new plans had to be "compatible" with the existing withdrawal agreement, which has been rejected three times by MPs.
There was then confusion after Mr Bettel held a press conference without Mr Johnson amid noisy protests by anti-Brexit protesters.
Mr Bettel, who addressed the media on his own after the UK PM pulled out, said his counterpart "holds the future of all UK citizens in his hands" and suggested it was his responsibility to break the deadlock in the process.
Standing next to an empty lectern, Mr Bettel warned Mr Johnson "you can't hold the future hostage for party political gain".
He said there were "no concrete proposals at the moment on the table" on a new Brexit deal from the UK and said the EU "needs more than just words".
"We need written proposals and the time is ticking, so stop speaking and act," he said.
The existing withdrawal agreement was the "only solution", he added.
Mr Johnson said his joint press conference was cancelled over fears the two leaders would have been "drowned out" by pro-EU protesters.
It is understood that his request for it to be held inside was turned down.
What exactly should we make of the oh so public venting on Monday by the prime minister of Luxembourg following his meeting with Boris Johnson?
Does this mean the EU has lost patience and will no longer engage in negotiations with the Johnson government? Can we expect an Angela Merkel rant or a Mark Rutte rave next?
Quite the opposite.
"As long as there is a chance of a deal, it's in our own interest to engage. However frustrating negotiations are," a high-level EU contact told me.
The EU's Brexit co-ordinator, Guy Verhofstadt, tweeted a photograph of the empty podium where Mr Johnson had been due to speak alongside Mr Bettel with the caption: "From Incredible Hulk to incredible sulk".
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.
After the working lunch with Mr Juncker, Mr Johnson told the BBC's political editor he was "cautiously optimistic" about the state of negotiations and suggested the EU wanted to bring the two and half years of arguments about the terms of the UK's exit to an end.
"I see no point whatever in staying on in the EU beyond October 31st and we're going to come out. And actually that is what our friends and partners in the EU would like too.
"And I think that they've had a bellyful of all this stuff. You know they want to develop a new relationship with the UK. They're fed up with these endless negotiations, endless delays."
While he was working "very hard" to get a deal, Mr Johnson said there would be no agreement unless the EU shifted its position on the backstop, the insurance policy to maintain an open border on the island of Ireland unless and until another solution is found.
"If we can't get movement from them on that crucial issue... we won't be able to get it through the House of Commons, no way."
He said there were a number of ideas under discussion which would allow the whole of the UK to leave the EU while protecting the integrity of the bloc's single market, upholding the Good Friday Agreement and supporting the Irish economy.
'Load of claptrap'
These, he said, included the use of technology to minimise border checks as well as the so-called Stormont lock, a mechanism to give Northern Irish politicians a say on the rules that apply to Northern Ireland.
"It is all doable with energy and goodwill," he insisted.
A UK government source later said: "It's clear Brussels is not yet ready to find the compromises required for a deal, so no-deal remains a real possibility - as the gap we need to bridge remains quite large."
As soon as we arrived at the office of the prime minister of Luxembourg it became obvious a planned outdoor news conference could not go ahead.
The anti-Brexit protesters in the square numbered less than 100 but their music and megaphones made it sound like a lot more and they occasionally used language you wouldn't want to hear on the news.
Behind the scenes the British and Luxembourgish delegations grappled with a diplomatic dilemma: Move the event inside but exclude the majority of the journalists? Gamble that the demonstrators could pipe down for a bit? Silence the host to save the guest's blushes?
The end result saw Mr Johnson do a short interview at the ambassador's residence to be shared with everyone while Mr Bettel took to the stage next to an empty podium.
He used the moment in the spotlight to deliver an impassioned speech, made all the more dramatic by the fact he's famed as one of the EU's most smiley, mild-mannered leaders.
Mr Johnson said he would meet the Halloween Brexit deadline come what may, insisting that the UK would be "in very good shape" whether there was a deal or not.
But pushed on how he would get around the law requiring him to ask for an extension if there is no deal by 19 October, the PM did not explain how it would be possible.
Ahead of Tuesday's Supreme Court hearing into whether the prorogation of Parliament was lawful, Mr Johnson defended the decision to suspend Parliament.
Parliament was prorogued last week, ahead of a Queen's Speech on 14 October. Legal challenges to the decision have been lodged in the courts by opposition MPs and campaigners.
Mr Johnson described claims that Parliament was "being deprived of the opportunity to scrutinise Brexit" as "all this mumbo jumbo" and a "load of claptrap".
"I think people think that we've somehow stopped Parliament from scrutinising Brexit.
"What absolute nonsense. Parliament will be able to scrutinise the deal that I hope we will be able to do both before and after the European Council on October 17."