Boris Johnson and Irish premier Leo Varadkar have said they can "see a pathway" to a possible Brexit deal.
Following more than two hours of talks at a country manor on the Wirral, the two leaders said they believed a deal was "in everybody's interests".
In a joint statement, they said they would now "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.
"They agreed to reflect further on their discussions and that officials would continue to engage intensively on them."
Mr Johnson and the Taoiseach released the joint statement after leaving talks at Thornton Manor this afternoon.
The UK Prime Minister's Twitter account said: "The Prime Minister and Taoiseach have had a detailed and constructive discussion ... they also discussed the potential to strengthen bilateral relations, including on Northern Ireland."
The statement added: "Following their discussions the Taoiseach will consult with the Taskforce 50 and the Brexit Secretary Stephen Barclay will meet Michel Barnier tomorrow morning."
The cautiously 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 an agreement in place for EU leaders to sign off at their summit on 17-18 October that would enable Mr Johnson to take Britain out of the EU on 31 October 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.
And he said Mr Johnson's proposals for a trusted traders scheme, with any physical customs checks taking place 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 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 19 October.
Many 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.