Boris Johnson will tell EU leaders there needs to be a new Brexit deal when he makes his first trip abroad as PM later this week.
The UK will leave the EU on 31 October with or without a deal, he will insist.
Meanwhile, the Sunday Times has printed leaked government documents warning of food, medicine and fuel shortages in a no-deal scenario.
A No 10 source told the BBC a former minister leaked the dossier to try to influence discussions with EU leaders.
The documents say the cross-government paper on preparations for a no-deal Brexit, codenamed Operation Yellowhammer, reveals the UK could face months of disruption at its ports.
It also states plans to avoid a hard border between Northern Ireland and the Republic of Ireland are unlikely to prove sustainable.
The dossier, reported by the Sunday Times, says leaving the EU without a deal could lead to:
The Downing Street source told the BBC the leaked document "is from when ministers were blocking what needed to be done to get ready to leave and the funds were not available".
Michael Gove, who is responsible for overseeing the devolution consequences of Brexit, said in a tweet that Operation Yellowhammer was "a worst case scenario".
"V significant steps have been taken in the last 3 weeks to accelerate Brexit planning," he added.
Energy Minister Kwasi Kwarteng told Sky News' Sophy Ridge on Sunday: "I think there's a lot of scaremongering around and a lot of people are playing into project fear."
But a former head of the British civil service, Lord Bob Kerslake, who described the document as "credible", said the dossier "lays bare the scale of the risks we are facing with no-deal Brexit in almost every area".
"These risks are completely insane for this country to be taking and we have to explore every avenue to avoid them," he told BBC Radio 4's Broadcasting House.
Irish deputy prime minister Simon Coveney said, in a tweet, that Ireland had "always been clear" a hard border in Ireland "must be avoided".
The Irish backstop - the provision in Theresa May's withdrawal agreement that could see Northern Ireland continue to follow some of the same trade rules as the Republic of Ireland and the rest of the EU, thus preventing a hard border - was an "insurance policy" designed to protect the peace process, he said.
Liberal Democrat MP Tom Brake said the leaked documents showed the effects of a no-deal Brexit should be taken more seriously.
"The government have simply, I think, pretended that this wasn't an issue," he said
The government was in "a real pickle", since the "the US has said that if that border is jeopardised, we're not going to get a trade deal with them", he said.
Speaker of the US House of Representatives Nancy Pelosi said, on Wednesday, a US-UK trade deal would not get through Congress if Brexit undermined the Good Friday Agreement.
The leak comes as the prime minister prepares to travel to Berlin to meet German Chancellor Angela Merkel on Wednesday, before going to Paris to meet French President Emmanuel Macron on Thursday.
Mr Johnson is expected to say Parliament cannot and will not change the outcome of the 2016 referendum and insist there must be a new deal to replace Mrs May's withdrawal agreement - defeated three times by MPs - if the UK is to leave the EU with a deal.
However, it is thought their discussions will chiefly focus on issues such as foreign policy, security, trade and the environment, ahead of the G7 summit next weekend.
Boris Johnson had been reluctant to fly to meet European leaders until it seemed a breakthrough was likely.
But - it still doesn't.
When Mr Johnson meets the EU's most powerful leaders - Angela Merkel and Emmanuel Macron - he will repeat his message that the UK is leaving, no matter what, at the end of October.
He will tell them face-to-face for the first time that the only way the UK will sign up to a deal is if the EU thinks again, and replaces the agreement brokered by Mrs May.
But there seems to be little chance of any serious progress in the coming days.
No 10 does not seem particularly optimistic and says it expects both sides will say their piece, then move on to other issues.
Anti-Brexit campaigner Gina Miller said the Government had "unequivocally" accepted it could not shut down Parliament to clear the way for a no-deal Brexit.
She told Sky's Sophy Ridge On Sunday: "What they have said is, unequivocally, they accept that to close down Parliament, to bypass them in terms of Brexit - stopping a no-deal Brexit, in particular - is illegal."
But Ms Miller said she would continue to seek further reassurances that MPs would be able to pass legislation to stop a no-deal Brexit.
Meanwhile, a cross-party group of more than 100 MPs has urged the prime minister to recall Parliament and let it sit permanently until the UK leaves the EU.
In a letter, MPs say the country is "on the brink of an economic crisis".
It continues: "Parliament must be recalled now in August and sit permanently until 31 October, so that the voices of the people can be heard, and that there can be proper scrutiny of your government."
Labour leader Jeremy Corbyn has reiterated his call for MPs to work together to stop a no-deal Brexit.
Speaking to the Observer, Mr Corbyn said his plan to be installed as an interim prime minister was the "simplest and most democratic way to stop no deal".
The Labour leader has said, as a caretaker PM, he would delay Brexit, call a snap election, and campaign for another referendum.
But Lib Dem leader Jo Swinson said Mr Corbyn was "divisive" and instead suggested Conservative MP Ken Clarke or former Labour leader Harriet Harman could head a temporary government.
Elsewhere, in a letter seen by the Mail on Sunday, Mr Johnson warned rebel Tory MPs their opposition to a no-deal Brexit was damaging the prospect of getting a new deal.
He said it was "plain as a pikestaff" that the EU will "not compromise as long as they believe there is the faintest possibility that Parliament can block Brexit on 31 October".