Boris Johnson is to meet Emmanuel Macron shortly, hours after the French president insisted reopening Brexit negotiations was “not an option”.
Mr Macron said: “We have to help the British deal with this internal democratic crisis but we mustn’t be hostage to it nor export it.”
On Wednesday, German Chancellor Angela Merkel said the onus was on the UK to find a workable plan.
Mr Johnson said a deal could be achieved with “sufficient patience”.
The UK prime minister has said the backstop – which aims to prevent a hard border on the island of Ireland after Brexit – must be ditched if a no-deal exit from the EU is to be avoided.
If implemented, the backstop would see Northern Ireland staying aligned to some rules of the EU single market, should the UK and the EU not agree a trade deal after Brexit.
It would also see the UK stay in a single customs territory with the EU, and align with current and future EU rules on competition and state aid.
These arrangements would apply until both the EU and UK agreed they were no longer necessary. Brexit supporters fear this could leave the UK tied to the EU indefinitely.
The EU has repeatedly said the withdrawal deal negotiated by former PM Theresa May, which includes the backstop, cannot be renegotiated.
However, it has previously said it would be willing to “improve” the political declaration – the document that sets out the UK’s future relationship with the EU.
Mrs Merkel has argued that the withdrawal agreement does not need to be reopened if a practical solution to the backstop can be found.
At a news conference in Berlin with Mr Johnson on Wednesday, she indicated that a solution to the backstop – a key Brexit sticking point – might be achievable within 30 days.
Mr Johnson responded that the German chancellor had set “a very blistering timetable of 30 days” adding: “I am more than happy with that.”
“I think that if we approach this with sufficient patience and optimism we can get this done and it is in the final furlong generally when the horses change places and the winning deal appears,” he added.
Later, Mr Macron said “Renegotiation of the terms currently proposed by the British is not an option that exists, and that has always been made clear by [European Council] President Tusk.”