Johnson said that there was a "pathway" to a belated deal to stave off a chaotic and costly no-deal Brexit on October 31, while Varadkar said the meeting was "very positive". Technical talks on an agreement to govern Britain's EU divorce are stalled, just days before what had been billed as a last chance European summit.
On Friday EU Brexit negotiator Michel Barnier said he had had a "constructive" meeting with his British counterpart, Stephen Barclay, following the Varadkar / Johnson meeting.
The pair are attempting to strike a deal before the proposed Brexit date of October 31.
If no deal is obtained next week, the sitting would also mark a crunch decision on whether Johnson agrees to request a three-month extension to the Article 50 process or is forced to resign.More news: Things Get Weird in New Trailer for Pixar's Onward
Negotiators have been given the green light for talks to enter the so-called "tunnel" phase - diplomatic jargon for talks continuing in secret between a small group of negotiators in complete secrecy - with a view to presenting a deal ahead of the European Union summit on 17 and 18 October.
"We are working towards a deal", said Mina Andreeva, spokeswoman for the European Commission. But Varadkar appears to be looking slightly further forward, implying that he and Johnson are now aiming for a deal in the next three weeks.
"I think it's possible for us to come to an agreement, to have a treaty agreed to allow the United Kingdom to leave the European Union in an orderly fashion by the end of October", he said.
"There's many a slip between cup and lip", he warned.
Dublin will have to consent to any solution to the hardest Brexit riddle of all: how to prevent the British province of Northern Ireland becoming a backdoor into the EU's markets without having controls at the border.More news: Tesla Alerts Owners To Fully Charge Ahead Of Massive California Power Shutoff
Since the 2016 British referendum vote for Brexit, negotiations have bounced back and forth between London and Brussels.
"If Barclay delivers the detail of what Johnson discussed with Varadkar yesterday, we may go into more intense talks over the weekend", one senior European Union diplomat said. After British MPs rejected a "backstop" plan in a previous withdrawal agreement to keep the border open, Johnson presented a new proposal last week.
With the United Kingdom due to leave the world's biggest trading bloc on October 31, Brexit remains deeply uncertain; it remains unclear on what terms it will leave or indeed whether it will leave at all. But Brussels is adamant it will not agree to any plan that undermines the single market - which allows free movement of goods across Europe - or exacerbates tensions on the island of Ireland.
The Irish border has been largely invisible since British army checkpoints were taken down after a 1998 peace deal largely ended the violence between members of the region's pro-British majority and an Irish nationalist minority.More news: Burr: Russia is waging an information warfare campaign against the U.S.
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