Additional reporting by the Digital Desk and Press Association
Latest: The Taoiseach is confident that a Brexit deal can now be reached before the October deadline after meeting with Boris Johnson.
Leo Varadkar described the three-hour meeting with his British counterpart as "very positive and very promising" and a new round of negotiations will begin in Brussels tomorrow.
Speaking to reporters in Liverpool airport, Mr Varadkar said: "We agreed not to get into the details of any task or discussions.
"But on the timeline question, I think it is possible for us to come to an agreement to have a treaty agreed to allow the UK to leave the EU in orderly fashion and to have that done by the end of October.
The Taoiseach says he believes it is now possible to come to an agreement before the end of Oct, he adds: “It remains our position that there can’t be a hard border between North and South” #iestaff pic.twitter.com/g0WwlZGGbC— Elaine Loughlin (@Elaine_Loughlin) October 10, 2019
"But there's many a slip between cup and lip and lots of things that are not in my control."
Mr Varadkar added: "It remains our position that there can't be a hard border between North and South.
"We must continue to have a situation whereby the all-island economy can continue to deepen and function well."
Former Taoiseach Bertie Ahern says there appears to have been a major breakthrough in today's talks.
"This is a total change around from where things were yesterday where we had Michel Barnier making it absolutely clear that what had been put to him by the British was not acceptable and the signals from Dublin were fairly negative all week," said Mr Ahern.
"Now today after the meeting there appears to be a substantive breakthrough."
Fianna Fáil's Brexit spokesperson Lisa Chambers has also said that the mood appears more positive this evening.
"If Boris Johnson can get a deal agreed with his cabinet, can he get it through the House of Commons because the arithmetic in the House of Commons has not changed for Boris it is the same number that Theresa May had," said Ms Chambers.
"I think for us we need to wait and see the details of what was offered today."
Update 3.30pm: Taoiseach Leo Varadkar and Boris Johnson have emerged from their meeting after talking for over three hours.
In a joint statement, the pair said that they have had a "detailed and constructive discussion".
The two leaders believe that reaching a deal is in everybody's interest and agreed that they could "see a pathway to a possible deal".
Among the things discussed in this afternoon's meeting were the challenges of customs and consent and the potential to strengthen bilateral relations, including on Northern Ireland.
Both Mr Varadkar and Mr Johnson have agreed to "reflect further" on their discussions and officials will continue to engage "intensively" on them.
The Taoiseach will consult with the EU Task Force regarding today's meeting while the UK Brexit Secretary Stephen Barclay will meet with EU chief negotiator Michel Barnier tomorrow morning.
The two leaders are still in Thornton House in Merseyside despite the joint statement having been issued.
The cautiously upbeat statement comes at the end of a week marked by acrimonious exchanges between Dublin, London 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 October 17 and 18 which would enable Mr Johnson to take Britain out of the EU on October 31 with a deal.
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 British Prime Minister has said that while he will abide by the law, he is determined to leave on the Halloween deadline come what may.
UK government sources have said ministers are preparing to hold an emergency Saturday sitting of Parliament on October 19.
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.
Update 1pm: The Taoiseach is currently meeting UK prime minister Boris Johnson outside Liverpool as Brexit enters a critical phase.
Mr Johnson arrived at Thornton Manor at 11.45am after flying by helicopter from London for the secret meetings.
Mr Varadkar was seen arriving to the Grade 2 Elizabethan property more than 20 minutes later.
The meeting’s location had been a closely guarded secret by both sides, with the Irish Government stating this was at the request of 10 Downing Street.
However, a small number of mainly Irish journalists were outside the Manor ahead of the meetings.
With the crucial EU summit in Brussels starting in a week, the chance of the Johnson securing a new Withdrawal Agreement is looking increasingly unlikely.
The location of the leaders’ lunchtime meeting has not been disclosed, other than that it is in the North West of England, and media coverage has not been invited.
The Taoiseach acknowledged ahead of the discussion that it will be “very difficult” to secure a deal by next week.
Mr Johnson wants to keep Northern Ireland more closely aligned with the EU than the rest of the UK on rules on goods and agriculture but remove it from the current customs union.
But Mr Varadkar is opposed to the Republic being in a different customs union from the north.
On Wednesday, EU chief negotiator Michel Barnier said there was still no basis for a fresh agreement on the UK’s withdrawal from the the bloc.
He said they had yet to see any “operational, legally binding solution” to the issue of the backstop ahead of next week’s European Council meeting.
Mr Johnson’s “two borders” proposals, he said, were based on a system “that hasn’t been properly developed, that hasn’t been tested”.
The downbeat assessment from Mr Barnier was echoed by the Taoiseach, who said the PM was installing an obstacle to progress by insisting that Northern Ireland must leave the customs union with the rest of the UK.
“That’s their position at the moment and that’s one that is a great difficulty for us,” Mr Varadkar told the Irish Parliament.
“As far as the Irish Government is concerned, we do want a deal, we’re willing to work hard to get a deal, to work until the last moment to get a deal, but certainly not at any cost.”
Mr Johnson must bring back a deal before October 19 if he is to avoid a clash over the Benn Act, which aims to prevent a no-deal departure.
The legislation orders the PM to ask for a delay to Article 50 until the end of January if MPs do not approve a deal before that date.
But he has repeatedly said he will not ask for a delay, while insisting that he will abide by the law.
Mr Johnson is planning an emergency Saturday sitting of Parliament following the summit, to be held on October 19, according to Government sources.
It is thought the could use the occasion to force a showdown with MPs determined to block a no-deal Brexit.