The European Union's Brexit negotiator Michel Barnier has said it is difficult to see a way to break the impasse as British Prime Minister Boris Johnson's demand to drop the backstop element of the withdrawal deal is unacceptable.
Hopes of a deal to ease the transition were stoked when Mr Johnson said the shape of an accord was emerging and European Commission President Jean-Claude Juncker said an agreement was possible.
However Mr Barnier has cast doubt on the likelihood of a deal, and reaffirmed that the bloc could not agree to London's demand to remove the backstop - the policy to prevent a return of border controls on the island of Ireland - without a serious alternative.
"I am sure you understand this is unacceptable," Mr Barnier said during a news conference in Berlin alongside German Foreign Minister Heiko Maas.
EU sources have said no proper alternative for the border has been proposed yet by London, so no breakthrough is on the cards.
In a quip about British talk of virtual checks on the border, Mr Barnier said: "I don't know how to inspect a cow with virtual methods."
"Based on current UK thinking, it is difficult to see how we can arrive at a legally operative solution which fulfils all the objectives of the backstop. It is in a very sensitive and difficult phase," he said.
Mr Johnson has vowed to take the UK out of the EU by the current deadline of 31 October, with or without a transition deal.
With the outcome of the marathon Brexit process still mired in doubt, the United Kingdom Supreme Court will rule tomorrow whether Mr Johnson's decision to suspend parliament was unlawful.
If the judges rule against the government at 10.30am, Mr Johnson could be forced to recall parliament - a step that widens the scope for MPs to block his "do-or-die" Brexit plans.
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Mr Johnson has said he wants to secure an amended deal at an EU summit on 17 October, and that "a large number of the important players", including Britain, Germany, France and Ireland, are keen to reach an agreement.
He said it was important the UK "whole and entire" was able to break away from EU law in future.
"The problem with ... the current backstop is that it would prevent the UK from diverging over a huge range of industrial standards and others," he said. "We may want to regulate differently but clearly there is also a strong incentive to keep goods moving fluidly, and we think we can do both."
An EU diplomat has said that Mr Johnson's government was sticking to a counter-productive approach imposing "its own, self-made problems on us and demand that we fix them.
"...Why would we bend our rules for a country that is leaving? Why put ourselves at risk for a third country? Why help smooth their departure at our own cost?" the diplomat said.
Johnson plays down Brexit 'breakthrough' chances in NY
Meanwhile, Mr Johnson has played down the chances of a "New York breakthrough" on Brexit during discussions with key EU leaders at the United Nations.
Mr Johnson will this week discuss his proposals for a new Brexit deal as the 31 October deadline looms.
He celebrated a "great deal of progress" having been made but sought to limit expectations when speaking to reporters on board the RAF Voyager on the way to New York's JFK airport.
"There will be discussions about Brexit. I would caution you all not to think that this is not going to be the moment... there will be no New York..." he said.
"There might be, but I don't wish to elevate excessively the belief that there will be a New York breakthrough.
"We will be pushing ahead but there is still work to be done."
He said it was "very encouraging" to hear Mr Juncker say he is not wedded to the mechanisms of the backstop.
Mr Johnson also said he had "seen interest" in treating the whole island as a single zone for sanitary and phytosanitary purposes, but warned it was "early days" and "there are clearly still gaps and still difficulties".
And on his current outlook, he said: "It would be still fair to say I'm in the same position I was. I think cautiously optimistic would be about right."
He will meet European Council president Mr Tusk today, before meeting German Chancellor Ms Merkel and French President Mr Macron.