The call happened on September 24, the source said, after French President Emmanuel Macron had shuttled between the US and Iranian leaders in a bid to arrange a historic encounter that he hoped would reduce the risk of all-out war in the Middle East.
"In New York, up to the last moment, Emmanuel Macron tried to broker contact, as his talks with presidents Trump and Rouhani led him to think contact was possible," the diplomatic source said.
Iran warns world to prepare for US nuke deal withdrawal
Speculation was abuzz last month that the leaders could meet on the sidelines of the General Assembly.
But Rouhani said he would only hold talks with the US if Trump lifted economic sanctions on Tehran.
Macron used his 48 hours in New York to see Trump three times and Rouhani twice, urging them to engage directly.
The source said Macron made a last-ditch attempt before flying back to Paris, with French technicians installing a secure phone line linking Trump's Lotte hotel and the Millennium, hosting the Iranian delegation.
The plan involved Trump calling at 9pm despite doubts over the Iranian reaction.
Macron went to the Millennium to ensure the phone call took place. Trump made the call, but Rouhani informed the French president he would not take it, the source said.
"The discussion continued to founder on the Iranians first wanting US sanctions lifted. Donald Trump wants Iran first to make commitments on its nuclear (ambitions) and ballistic and regional activities," the source said.
Iran says defence capabilities not negotiable amid US pressure
The French diplomatic source comments come after US reports emerged earlier this week about Macron's initiative to get the leaders to talk.
Tensions have been escalating between Iran and the United States since May last year when President Donald Trump pulled out of the nuclear accord and began reimposing sanctions that have crippled the Iranian economy.
Britain, France and Germany have repeatedly said they are committed to saving the deal that gave Iran relief from sanctions in exchange for curbs on its nuclear programme, but their efforts have so far borne little fruit.
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