Video report by ITV News Europe Editor James Mates
Donald Tusk has said he has received "promising signals" from Taoiseach Leo Varadkar that a Brexit deal is possible.
Mr Tusk, President of the European Council, tweeted in the wake of Thursday's meeting between the Irish premier and Boris Johnson on the Wirral.
Mr Tusk said: "The UK has still not come forward with a workable, realistic proposal. But I have received promising signals from Taoiseach @LeoVaradkar that a deal is possible.
"Even the slightest chance must be used. A no deal #Brexit will never be the choice of the EU."
Mr Tusk's comments came prior to Brexit Secretary Stephen Barclay holding key talks with the EU’s chief negotiator Michel Barnier on Friday as the chance of a breakthrough on securing a withdrawal deal appeared to rise.
Mr Barnier welcomed the "constructive meeting" with Mr Barclay.
After leaving the Brussels meeting, Mr Barnier told reporters: "We had a constructive meeting with Steve Barclay and the British team and now I'm going to debrief the 27 ambassadors and the Brexit steering group of the parliament.
"I've already said that Brexit is like climbing a mounting: we need vigilance, determination and patience."
Michel Barnier says his meeting with Stephen Barclay was 'constructive'
The Brussels meeting follows discussions between Boris Johnson and Mr Varadkar after which both leaders declared they could “see a pathway” to a possible agreement.
Mr Johnson said on Friday: "I had a good conversation with the Irish Taoiseach Leo Varadkar yesterday [Thursday] and I think both of us can see a pathway to a deal but that doesn't mean it's a done deal - there's a way to go."
Mr Johnson also discussed the prospect of Northern Ireland staying in the Customs Union.
“I think it would be wrong of me to give a running commentary on the negotiations,” he said. “With the greatest possible respect I think, look at everything I’ve said previously. I think you can draw your own conclusions from that. But let’s our negotiators get on.”
Johnson added: “I can certainly tell you that under no circumstances will we see anything that damages the ability of the whole of the United Kingdom to take full advantage of Brexit, and I think that’s what people would expect, and that’s what I think we can achieve.”
Mr Barclay’s talks look likely to give the clearest indication of whether there has been a significant thawing of relations between London and the EU after a week of abrupt exchanges.
Such a move could lead to the start of so-called intensive “tunnel” negotiations in the coming days ahead of the crunch EU heads of government summit on October 17 and 18.
The unexpectedly cordial atmosphere of the meeting between the two prime ministers led to speculation of a possible compromise on the contentious issue of the Northern Ireland border backstop.
Sterling rose sharply on international money markets in the wake of the talks.
ITV News Europe Editor James Mates provides the latest updates on Brexit talks
After more than two hours of “detailed and constructive” discussions at a country manor on the Wirral on Thursday, the two leaders said it was in “everybody’s interest” to get an agreement which would allow the UK to leave with a deal.
Mr Varadkar said he hoped the progress they had made would be “sufficient” to enable intensive negotiations to resume in Brussels ahead of next week’s crucial EU summit.
The Taoiseach said: “I think it is possible for us to come to an agreement, to have a treaty to allow the UK to leave the EU in an orderly fashion, and to have that done by the end of October, but there’s many a slip between cup and lip,” he said.
“In terms of how long it will take, I can’t predict that with any certainty, but I think all sides would like there to be an agreement next week at the council if possible.
“Obviously there’s a further deadline after that which is the 31st of October, so I would say a short pathway rather than a long one, but it’s impossible to predict that for sure.”
Mr Varadkar refused to be drawn on any “concessions” made by either side, while UK Government sources refused to be drawn on Irish press reports suggesting “significant movement” by the UK.
Tory former chancellor and arch-Brexiteer Lord Norman Lamont said he wanted to know what any UK concessions were.
He told the BBC: “No, I am not worried. I have to know what the concessions are.
“It does look as though there is a change of mood.”
He added: “Maybe there is some change on the mechanism whereby the deal is approved by the Northern Ireland Assembly and Executive.
“Maybe there will be an opting-out mechanism rather than an opting-in mechanism.”
The softer mood music after Mr Johnson’s meeting with the Taoiseach followed an intensive few days which saw an acrimonious war of words explode between London, Dublin and Brussels in which the talks appeared close to collapse.
Briefings by anonymous Downing Street sources had accused Mr Varadkar of backtracking on previous commitments to try to find a deal, and of refusing to negotiate.
And following a heated telephone call between Mr Johnson and German Chancellor Angela Merkel on Tuesday, Number Ten sources claimed the EU was making it “essentially impossible” for Britain to leave with a deal.
On Wednesday, Mr Barnier told the European Parliament there was still no basis for a fresh agreement.
He said the UK had yet to put forward an “operational, legally binding solution” to replace the Northern Ireland backstop – intended to prevent the return of a hard border with the Republic.
Without a deal, Mr Johnson will face demands from opposition parties to comply with the so-called Benn Act which would require him to request a three month Brexit delay if there is no agreement by October 19.
The PM has said while he will abide by the law, he is determined to leave on the Halloween deadline of October 31 come what may.
Government sources have said ministers are preparing to hold an emergency Saturday sitting of Parliament on October 19.
Many MPs believe if he cannot get a deal, Mr Johnson will use the occasion to push for a “people versus Parliament” general election, possibly as early as next month.
If a deal did emerge, Mr Johnson would also need the backing of the DUP and Eurosceptic Tories to have any chance of getting it through without opposition support.
Lord Lamont said he thought Labour MPs were wary of an early election.
He said: “Backbench Labour MPs, I understand, are really very nervous about it.
“I think they don’t want it, and I wouldn’t be at all surprised if we didn’t slide into next year.”
Lord Lamont said the Tories would choose to fight any election on a no deal stance if they failed to achieve an agreement with Brussels.