The prime minister and his Irish counterpart are conversing at Thornton Manner in Wirral, which describes itself as a "luxury" wedding venue.
They are holding what both sides described as a "private meeting", in order to allow them and their teams to have detailed discussions.
They will meet after EU chief Brexit negotiator Michel Barnier stated the UK and the bloc are "not really in a position where we are able to find an agreement".
Mr Johnson has unveiled new plans for the post-Brexit Irish border, having vowed to scrap the backstop arrangement agreed between Brussels and his predecessor Theresa May.
Mr Barnier has said the Irish border proposals are "not something we can accept", while Mr Varadkar has also been cool on the plans.
He told the Irish parliament on Wednesday that Mr Johnson's insistence that Northern Ireland must leave the EU's customs union with the rest of the UK was blocking progress in reaching an agreement.
"That's their position at the moment and that's one that is a great difficulty for us," Mr Varadkar said.
"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."
The backstop is designed as an insurance mechanism to prevent a hard border on the island of Ireland regardless of the future EU/UK trade relationship.
Mr Johnson has branded the arrangement "undemocratic", while Brexiteers fear it could leave the UK trapped in the EU's customs union - limiting the capacity for new independent trade deals - as well as following EU rules but with no influence over them.
Business Secretary Andrea Leadsom reiterated on Wednesday night that the government "would not entertain a backstop", even if it was of a time-limited nature.
The prime minister hopes to strike a deal ahead of next week's summit of EU leaders, and before the 31 October Brexit deadline - the date on which he has vowed to take the UK out of the EU "do or die".
Mr Johnson has repeatedly stated he will not delay Brexit beyond 31 October, but legislation passed by MPs will compel the prime minister to seek an extension to the Article 50 negotiating if he does not secure a Brexit deal - or MPs explicitly approve a no-deal Brexit - by 19 October.
Ms Leadsom, speaking to ITV's Peston show, said it would be "perfectly reasonable" to stress to the EU that the government does not want a delay.
It follows suggestions the prime minister could write one letter asking for an extension, in a possible attempt to abide by the law, but then also write a second letter telling the EU that ministers don't want one.
Labour's shadow Brexit secretary Sir Keir Starmer said: "If there is no deal by the end of next week, the prime minister must ask for, and accept, an extension. One letter. No equivocation."
Mr Johnson's defeated rival for the Conservative leadership, Jeremy Hunt, made his first major Brexit intervention on Wednesday night to implore the EU not to make a "catastrophic miscalculation".
In a letter to EU foreign ministers, the ex-foreign secretary wrote: "I believe the Brexit outcome is about to be made worse - for both the EU and the UK.
"Once again, I fear a profound and mutual lack of understanding is leading the EU to make the same mistakes over and over again.
"I am hoping and praying that does not happen because the implications for our future relationship would be extremely grave."
Labour leader Jeremy Corbyn will on Thursday use a speech to repeat his demand for Mr Johnson to avert a no-deal Brexit on 31 October, before his party will support the prime minister's call for general election.
Parliament is currently suspended ahead of Monday's Queen's Speech, in which the government will set out its legislative agenda.
In a speech detailing his alternative programme, Mr Corbyn will say: "It wasn't long ago that Johnson was pretending not to want an election. Now he is pretending that it's Labour that doesn't want one.
"So let me address this directly: Prime minister, we can't trust you not to break the law because you've got form.
"We can't trust you not to use the period of an election campaign to drive our country off a no-deal cliff edge that will crash our economy, destroy jobs and industries, cause shortages of medicine and food and endanger peace in Northern Ireland.
"So it's simple: obey the law, take no-deal off the table and then let's have the election.
"We're ready and champing at the bit. There's only one reason it hasn't happened yet - we can't trust you."
Opposition parties last month twice defeated Mr Johnson's attempts to call a snap general election.
In an article in Thursday's Daily Telegraph, Brexit Party leader Nigel Farage reissued his offer of an election pact with the Mr Johnson's Conservatives.
He wrote: "Winning a huge Brexit majority in parliament is the prize that awaits.
"If we work together to fight against the real enemy, Jeremy Corbyn and his bunch of wreckers, we would deliver a positive campaign with a decisive result."