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 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.
Instead, 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.
The talk comes as The Sunday Times published leaked government documents on 'Operation Yellowhammer', in which it warned of a three-month "meltdown" at its ports in the event of a no-deal Brexit.
The paper also warned of food, medicine and fuel shortages and a hard Irish border.
A senior Whitehall source told the paper: "This is not Project Fear - this is the most realistic assessment of what the public face with no deal.
"These are likely, basic, reasonable scenarios - not the worst case."
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.
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.
On a visit to Bolton, Mr Corbyn dodged the question of whether the likes of veterans Ken Clarke, of the Conservatives, or Labour's Harriet Harman would be less divisive figures to lead a unity government.
He said: "What we need is a respect for the electoral process that brought about the results from the last general election.
"What we need is a government that is prepared to negotiate with the European Union so we don’t have a crash-out on the 31st.
"This Government clearly doesn’t want to do that. Boris Johnson just wants to take us into the arms of the Americans and Donald Trump on a sweetheart trade deal.
"We are not going to do that. We will do everything we can to stop a no-deal Brexit.
"I am the leader of the Labour Party, Labour is the largest opposition party by far. That is the process that must be followed."
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.
Meanwhile, the Mail on Sunday reported 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".