Taoiseach Leo Varadkar and his British counterpart Boris Johnson have said they can "see a pathway" to a possible Brexit deal, breaking the deadlock on the Northern Ireland backstop.
Following more than two hours of "detailed and constructive" talks at a country manor on the Wirral, 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.
Brexit Secretary Stephen Barclay will meet the EU's chief negotiator, Michel Barnier, in Brussels today when they are expected to assess whether there are the grounds to move forward.
Mr Johnson almost certainly needs the EU leaders gathering in Brussels on October 17 and 18 to sign off on an agreement in order to be able to take Britain out of the EU on October 31 with a deal.
Speaking to reporters at Liverpool Airport before his return to Dublin, Mr Varadkar said while there were still issues to be resolved, he believed it was possible to meet the Halloween deadline.
"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."
Earlier in a joint statement, the two leaders said they would "reflect further" on their discussions while their officials would continue to "engage intensively".
"Both continue to believe a deal is in everybody's interest. They agreed that they could see a pathway to a possible deal," the statement said.
Mr Varadkar refused to be drawn on any "concessions" made by either side, while British sources refused to be drawn on Irish press reports suggesting "significant movement" by the UK.
The meeting at the 19th-century Thornton Manor - now a luxury wedding venue - was seen as a last chance for Mr Johnson to get his hopes of an agreement back on track.
In their statement, the two leaders said their discussions had concentrated on the "challenges" of future customs arrangements and "consent" in Mr Johnson's Brexit blueprint.
The Irish and other EU governments have objected to proposals to take Northern Ireland out of the EU customs union - along with the rest of the UK - meaning the return of customs checks on the island of Ireland.
They have also voiced strong concern about proposals in the plan for the new arrangements to require the consent of the Stormont Assembly, effectively handing a veto to the DUP.
The unexpectedly upbeat statement comes at the end of a week marked by acrimonious exchanges between London, Dublin and Brussels in which the negotiations appeared close to collapse.
Briefings by anonymous No 10 sources 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, they claimed the EU was making it "essentially impossible" for Britain to leave with a deal.
Time remains tight, however, if there is to be 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. And he said Mr Johnson's proposals for a trusted traders scheme, with any physical customs checks taking place well away from the border, were based on a system "that hasn't been properly developed, that hasn't been tested".
If there is no agreement, Mr Johnson will face demands from opposition parties to comply with the so-called Benn Act which would require him to go back to Brussels and request a further Brexit delay.
The Prime Minister has said that while he will abide by the law, he is determined to leave on the Halloween deadline come what may. Government sources have said ministers are preparing to hold an emergency Saturday sitting of Parliament on October 19.
Many British MPs believe that if he cannot get a deal, Mr Johnson will use the occasion to lambast them for thwarting an agreement, laying the ground for a "people versus Parliament" general election, possibly as early as next month.