Mr Johnson will make clear to the French president and German chancellor that Britain will leave the European Union on October 31 with or without a deal.
The PM, who is heading to Berlin on Wednesday and Paris on Thursday, is expected to say that Parliament will not and cannot cancel the outcome of the EU referendum.
He will insist there must be a new deal to replace Theresa May's thrice-defeated Withdrawal Agreement if Britain is to leave with a settlement on October 31.
However, Number 10 said it expects there will be "very little discussion" of Brexit during the visits, predicting that each side would state its position and then move on to other topics.
• READ MORE: Euan McColm: Why Corbyn’s plan to bounce Boris came to griefInstead, it is thought the discussions will revolve around next weekend's G7 agenda - with topics including foreign policy, security, trade and the environment likely to dominate.
Mr Johnson will meet world leaders at the summit in Biarritz, France, where he will seek to spread the message of the UK's "renewed global reach".
He is also expected to discuss how states can work together to address challenges facing the world's biggest economies, such as fears about the financial system, security issues and climate change.
European Commission president Jean-Claude Juncker, 64, who was due to attend the gathering, has had to cut short his summer holiday for an emergency gallbladder operation.
Details of the PM's travel plans emerged as Labour leader Jeremy Corbyn reiterated his call for MPs to work together to prevent a no-deal Brexit.
Mr Corbyn, who set out his plan to be installed as a caretaker prime minister last week to stop the UK crashing out of the EU without a deal, said his proposal is the "most democratic way" to prevent a no-deal.
He told the Observer: "My message to MPs across Parliament is simple and urgent: only by working together can we stop no-deal.
"Three years after the EU referendum, the country stands at a precipice. Boris Johnson has become Prime Minister without any popular mandate. He has no right to drive our country off a cliff and into the arms of Donald Trump with his no-deal fixation.
"The plan I set out this week is the simplest and most democratic way to stop no-deal. We have to seize the opportunity before it's too late, so the people, rather than an unelected Prime Minister, can decide our country's future."
The Liberal Democrats and senior Tories have rejected his proposal, however it won the potential backing of the SNP, Plaid Cymru and Tory MP Guto Bebb.
• READ MORE: Boris Johnson reveals just how much he cares about free speech and democracy – Joyce McMillanMeanwhile, the Mail on Sunday reported that Mr Johnson has accused former chancellor Philip Hammond of "gravely damaging" the national interest with his bids to frustrate Brexit.
In a letter seen by the paper, the PM said it was "plain as a pikestaff" that the EU "will simply not compromise as long as they believe there is the faintest possibility that Parliament can block Brexit on 31 October".
The Sunday Telegraph reported a Downing Street source accusing Tory Remainers of "appalling dishonesty" for attempting to use the Brexit delay to "cancel the referendum".
The paper also said that Brexit Secretary Stephen Barclay has signed the "commencement order" which will trigger the end of the supremacy of EU law in the UK on October 31.
Business Minister Kwasi Kwarteng played down suggestions that there could be an election campaign ahead of October 31.
"I don't think anyone is talking about calling an election, there's a lot of speculation about things like a government of national unity and all the rest of it," he told Sky's Sophy Ridge on Sunday.
"I don't see that as a way forward. In order to have a general election lots of things have to happen which, I think, in my opinion, are unlikely to happen but we'll have to wait and see."