The UK's proposals for the Irish border are still neither "workable" nor "realistic", the EU has said, following hours of last-ditch talks in Brussels on Friday aimed at striking a deal.

Brexit Secretary Steve Barclay had hoped to break months of deadlock at a breakfast meeting with his counterpart Michel Barnier in the EU capital.

But talks broke up after two hours with little to show, despite both sides describing them as "constructive". European Council president Donald Tusk said there had been "promising signals" and that a deal was still possible, but that the UK needed to move further.

"Unfortunately, we are still in a situation in which the UK has not come forward with a workable, realistic proposal," Mr Tusk said.

"A week ago I told PM Johnson that if there was no such proposal by today, I would announce publicly that there are no more chances – because of objective reasons – for a deal during the incoming European Council.

"However, yesterday when the Irish Taoiseach and the UK Prime Minister met they both saw – for the first time – a pathway to a deal. I have received promising signals from the Taoiseach that a deal is still possible."

He added that "technical talks" were still taking place in Brussels but that there was "no guarantee of success and the time is practically up".

The mood of talks swung towards last-minute optimism on Thursday after a meeting between Boris Johnson and Irish PM Leo Varadkar on Merseyside, where both sides spoke of a "pathway to a deal".

Officials are being tight-lipped about the content of discussions and is not clear what it was that Mr Johnson and Mr Varadkar discussed that has raised hopes of a solution.

Only the day before the EU's chief negotiator Michel Barnier had savaged UK plans in a public point-by-point deconstruction.

Mr Barnier said the EU had three concerns with UK proposals: that they did not prevent a customs border on the island of Ireland, that they included a veto for the Northern Ireland Assembly, and that they were not actually legally operable or ready to go.

Boris Johnson and Leo Varadkar's meeting on Thursday sparked optimism (PA)

A UK government spokesperson told reporters in Brussels: “Steve Barclay had a constructive meeting with Michel Barnier this morning. Michel Barnier will brief Ambassadors from the EU27 this afternoon.”

A European Commission spokesperson had much the same message, stating: "What I can say from our side is that Michel Barnier had a constructive meeting this morning with Steve Barclay and now he will debrief the EU27 ambassadors on COREPER and also inform the Brexit Steering Group accordingly."

She added that Commission negotiators had been in touch with Irish government representatives following the meeting between Mr Varadkar and Mr Barnier, "in order to be on the same page with regards to the latest developments".

The spokesperson decline to go into detail but told reporters that "you can assume that they exchanged ideas or they discussed many different angles".

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Michel Barnier himself said: “We have had a constructive meeting with Steve Barclay and the British team and now I am ready to debrief the 27 ambassadors and the Brexit steering group of the parliament. But as already said, Brexit is like climbing a mountain. We need vigilance, determination and patience.”