The Taoiseach's meeting will take place amid a growing sense in Europe that an extension is now the only viable option.
Johnson has vowed Britain will go away on October 31 with or and not using a deal - regardless of MPs passing a legislation final month that requires him to hunt one other Brexit delay if he fails to safe a pact on the summit. "Looking forward to a detailed discussion to see if we can make any progress", Varadkar said on Twitter, posting pictures of the two leaders sitting down to meet.
Mr Varadkar has expressed concern about Mr Johnson's proposal to give the Northern Ireland Assembly a vote over entering into a "regulatory zone" with the EU, which would involve it leaving the customs union.
"To put things very frankly, though, and to try and be objective, (at) this particular point, we are not really in a position where we are able to find an agreement", he said.
His comments came as Ms Leadsom hinted the PM was gearing up to by-pass legal obstacles to a no-deal Brexit by sending one letter requesting an extension and, in the same instance, submitting a second memo telling European leaders he does not.
"That creates huge difficulties for us because we want there to be a deal that respects the wishes of the people of Northern Ireland, and indeed the people in this Republic too".
Brexit supporters say this can be managed by doing checks on goods away from the border, but critics say it will be hard to police this without any physical infrastructure like border posts of cameras, which could raise tensions in the divided communities of Ireland.
"They're seriously focused on trying to resolve this issue and trying to get a deal", junior business minister Kwasi Kwarteng said.
The Prime Minister's "two borders" proposals, he said, were based on a system "that hasn't been properly developed, that hasn't been tested".More news: 'We too love money more than freedom'
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Delaying Brexit would only increase the chances of a no-deal exit, he warned.
Johnson's proposals crossed so numerous EU's red lines, Barnier said, that he would need to fundamentally revamp his proposals - something the British leader has said he will not do.
The Guardian reported that the European Union may offer to extend the Brexit deadline until June 2020. Johnson's legislative agenda will be set out by the British monarch in the traditional Queen's Speech. The speech will then be debated by MPs throughout the week.
The UK prime minister has until the European Council summit on October 17 and 18 to secure an agreement with Brussels.
European Parliament head David Sassoli said Britain now had a choice between a no-deal Brexit or a delay.
European officials warned Monday (6 October) that Britain's latest Brexit proposal can not serve as a basis for a breakthrough before next week's Brussels summit.
British parliament reconvenes, setting the stage for further attempts by lawmakers to ensure the October 31 deadline for withdrawal is extended, regardless of the prime minister's promise that the United Kingdom will leave on that date, deal or no deal.
The British parliament will hold a special Saturday sitting on October 19 to decide what to do.