France's Emmanuel Macron said on Monday that conditions were right after the G7 summit for a meeting between US President Donald Trump and Iranian leader Hassan Rouhani.
Mr Trump appeared to confirm his willingness at the G7 summit in Biarritz, south-west France, saying he would meet the Iranian leader if the situation was right.
"If the circumstances were correct, I would certainly agree to that," he said.
Mr Trump said progress could be made, although it "may not be immediately".
But he warned that if Iran failed to agree to being a "good player" it will be met with "very violent force".
Mr Macron said he hoped a summit could take place in the next few weeks.
"Nothing is for sure. Things are eminently fragile," he said.
"Two things are very important for us: Iran must never have nuclear weapons and this situation should never threaten regional stability."
Mr Macron said the Iranian president had told him that he was open to a meeting.
Mr Trump said the timeline proposed by Mr Macron was realistic and he believed Mr Rouhani would also be in favour, he added.
"I think he's going to want to meet," he said. "I think Iran wants to get this situation straightened out."
Mr Trump and Mr Rouhani are expected to be at the UN General Assembly in New York at the end of September.
Mr Trump said he knew Iranian Foreign Minister Javad Zarif would attend G7 talks.
He said the G7 countries had “great unity” in their talks about Tehran and appeared to leave the door open to diplomatic talks.
Mr Zarif, who has been sanctioned by the US, unexpectedly flew into France on Sunday.
The Iranian economy is struggling since the US reimposed sanctions after pulling out of a pivotal 2015 nuclear deal that put limits on Tehran's uranium enrichment programme.
Tension has soared between the two countries. European powers, who did not support the withdrawal of Washington from the 2015 agreement, have tried to mediate.
"I knew [Zarif] was coming in and I respected the fact that he was coming in," Mr Trump said.
"We're looking to make Iran rich again. Let them be rich, let them do well if they want."
Initial reports said the US president had been extremely angry over Mr Zarif's appearance in Biarritz and that Mr Macron had not warned Mr Trump of Iran's presence.
“He spoke to me, he asked me," Mr Trump said at a Monday morning meeting with Egyptian president Abdel Fattah El Sisi.
"I said if you want to do that, that’s OK. I don’t consider that disrespectful at all, especially when he asks me for approval."
Mr Zarif left the summit in France for China.
While Mr Trump said there had been a united stance on Iran, German Chancellor Angela Merkel said there was still some way to go.
But there was an agreement that Iran could not be allowed to obtain nuclear weapons.
"We've come to a conclusion, more or less. But the biggest part of the conclusion: they can't have nuclear weapons," Mr Trump said.
Mrs Merkel said the "firm will to talk" was a sign of progress and the support for a political solution was positive.
“There is an atmosphere at this summit in which talks are welcome, talks of Europe with Iran and especially of France with Iran,” she said.
“This happens in co-ordination with the US and that’s already a lot.”
Mrs Merkel did not reveal if Mr Trump was prepared to accommodate any move towards easing sanctions against Iran.
At the end of the summit, Mr Macron and Mr Trump together praised the event.
The French leader said there had been agreement on issues at the summit after three days of talks dominated by trade, the Iranian nuclear deal crisis and fires in the Amazon.
"We have managed to find real points of convergence – unprecedented, very positive – that will allow us to go forward in a very useful way," Mr Macron said.
Mr Trump said the French leader, with whom he has disagreed on policies towards Iran, had done a "fantastic job" at the G7, saying it had been a "very unified" meeting.
