Mr Johnson said: “At the moment it is absolutely true that our friends and partners are a bit negative… but I think actually we’ll get there.
“I think there is a real sense now that something needs to be done with this backstop. We can’t get it through parliament as it is.”
He added: “One thing that slightly 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’s a possibility that parliament will block Brexit, they’re unlikely to be minded to make the concessions that we need, so it’s going to take a bit of patience.”
Mr Johnson’s words are the latest in a tit-for-tat exchange with European Council President Donald Tusk over the backstop.
The backstop, which is intended to avoid the return of a hard border between Ireland and Northern Ireland if a UK-EU trade deal is not reached, has become a major stumbling block in Brexit talks.
“We think there’s a big opportunity now for everybody to come together, take out that backstop and then… in the course of the negotiations on the free trade deal… which we’re going to do after 31 October, we will be bringing forward all the ways in which we can maintain frictionless trade at the Northern Irish border,” the prime minister said.
These could include trusted-trader schemes and electronic pre-clearing to take checks away from the border, he added.
Mr Tusk had said that those opposed to the backstop were “not proposing realistic alternatives”.
He had been responding to Mr Johnson’s earlier insistence that the backstop is “anti-democratic” and that he is willing to give the EU the “commitments” needed for alternatives to be put into effect.
Mr Johnson had said he could not support any agreement that “locks the UK, potentially indefinitely, into an international treaty which will bind us into a customs union and which applies large areas of single market legislation in Northern Ireland”.
Meanwhile, it has been revealed that civil servants and ministers will now only attend EU gatherings where the UK has a significant national interest in the outcome of discussions, such as on security.
One serving UK diplomat told Sky News the move was counter-productive, saying: “The suggestion it will save us time is bonkers. We will now have to spend ages working out what happened in the meetings and how it affects us.”