The meeting was seen as a last chance for Mr Johnson to save his hopes of getting agreement on a Brexit deal ahead of next week’s crucial EU summit.
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 Northern Ireland’s Democratic Unionist Party. The statement said Varadkar will now consult with Brussels while Brexit Secretary Stephen Barclay will meet the EU’s chief negotiator Michel Barnier on Friday.
It comes as French President Emmanuel Macron warned Britain that Brexit was a “British domestic crisis” and it would have to take responsibility for how it leaves the European Union.
“Brexit is a British domestic crisis, not a European one,” Mr Macron said, adding that what was important now was to finalise negotiations and see if there was “something that I hope could fly.”
But he added: “At the very end this is a British responsibility”, on whether it leaves the EU with or without a deal or even cancels the Brexit process outright.
Mr Macron said it was time to resolve the issue as “we have already spent a lot of time” on discussing Brexit and the priority now was to discuss Britain’s future relationship with the EU.
“We have to prepare the future,” he said, speaking in English in the French city of Lyon.
Time is running out to sign off on any agreement at an October 17-18 EU summit, ahead of Britain’s scheduled departure from the bloc at the end of the month after nearly five decades of membership.