Mr Johnson then moves on to Paris for talks with President Emmanuel Macron on Thursday.
He will then hold further bilateral meetings with other leaders at the G7 summit in Biarritz this weekend.
On Monday evening, Mr Johnson wrote to EU Council President Donald Tusk setting out his proposed “next steps” to find a Brexit deal that removed the Irish backstop.
The prime minister claimed the commitment to keep Northern Ireland within the customs and regulatory orbit of the EU after Brexit was “undemocratic”, and requested it be replaced with a commitment to have “alternative arrangements” in place by the end of a two-year transition period.
But Mr Tusk dismissed the letter, saying the backstop was “an insurance to avoid a hard border on the island of Ireland”, and claimed those opposing it without presenting “realistic alternatives” were advocating the return of a hard border “even if they do not admit it”.
Ms Merkel also cast doubt on the prospects of the backstop being removed, during a visit to Iceland.
“The EU is ready to find a solution, but for that there is no need to re-open the withdrawal agreement. It is a question of future ties. Britain needs to decide which way it goes,” the German chancellor said.
It comes after US President Donald Trump claimed the EU “have not treated the UK very well” in the course of the Brexit negotiations, but said Mr Johnson was “the right man” to resolve the crisis.
Speaking to reporters in the Oval Office, he said “they are driving a very tough negotiation, the European Union, a very tough bargain”.
Mr Trump added: “I think the UK has the right man in charge right now, the right person in charge in the form of Boris Johnson.”
In a response last night, Boris Johnson claimed he remained confident a deal could be reached.
“At the moment it is absolutely true that our friends and partners are a bit negative.
“I saw what Donald Tusk had to say and it wasn’t relevant of a sense of optimism.
“But I think actually we will get there.”
He added: “I’m going to go at it with a lot of oomph, as you’d expect, and I hope we’ll be making some progress in the course of the next few weeks.
“But clearly, one thing that slightly I think complicates the picture is that our EU friends still clearly think that there is a possibility that parliament will block Brexit.
“As long as they think there is a possibility that parliament will block Brexit they are unlikely to make the concessions that we need, so it’s going to take a bit of patience,” he added.
But critics in parliament have suggested this is a misreading of the situation, and that the threat to leave without a deal on 31 October does not offer the diplomatic leverage the PM has suggested.
Speaking to Sky News, the chairman of the Brexit Select Committee, Hilary Benn said: “What does he do on 1 November? That is when the problems will begin.
“The moment he goes to Europe after a no-deal Brexit and says can we talk about a trade deal Europe will say, right well before we can do that, what about the money you owe and what about a backstop to guarantee an open border between Northern Ireland and the Republic? In other words, he’s going to have to address the things he refuses to address now”.
The government, however, insists the commitment to leave the European Union with or without a deal on 31 October remains unequivocal.
Yesterday Brexit Secretary Steve Barclay announced UK ministers and officials would stop attending most European Union meetings from 1 September.
Mr. Barclay said representatives would only be represented at meetings where the UK has a “significant national interest” in order to “free up time for ministers and their officials to get on with preparing for our departure on October 31st”.
Liberal Democrat Brexit spokesman Tom Brake said the decision was a “petulant move”.
A serving UK diplomat, speaking anonymously to Sky News, said the suggestion it would save officials time was “bonkers”.
“We will now have to spend ages working out what happened in the meetings and how it affects us,” they added.