Both Mrs Zaghari-Ratcliffe and fellow detainee Anoosheh Ashoori are believed to be heading to Tehran airport to leave the country.
Tulip Siddiq, Mrs Zaghari-Ratcliffe’s MP in Hampstead and Kilburn, wrote on Twitter: “Nazanin is at the airport in Tehran and on her way home.”
The Foreign Office has not commented on the reports and earlier Boris Johnson said negotiations about Mrs Zaghari-Ratcliffe were “moving forward” but “going right up to the wire”.
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There is still nervousness in Whitehall about the situation, with sources stressing the pair will not be free until they are actually on a plane out of Iran.
But the apparent breakthrough will bring an end to the ordeal for Mrs Zaghari-Ratcliffe, which began in 2016 when she was detained by the Iranian Revolutionary Guard at Imam Khomeini airport after a holiday visit to Iran where she introduced her daughter Gabriella to her parents.
There have been signs of progress in delicate negotiations between the UK and Iran in recent days.
But Mr Johnson, during a trip to the Middle East, was careful not to elaborate further when speaking to reporters on Tuesday
A glimmer of optimism for 43-year-old Mrs Zaghari-Ratcliffe came a day earlier when Ms Siddiq said her British passport had been returned.
Mr Johnson confirmed a British negotiating team was working in Tehran to secure the release of dual nationals, while Mrs Zaghari-Ratcliffe at the time remained at her family home in the Iranian capital.
“I really don’t think I should say much more, I’m sorry, although things are moving forward,” he told broadcasters at the Emirates Palace hotel in Abu Dhabi.
“I shouldn’t really say much more right now just because those negotiations continue to be under way and we’re going right up to the wire.”
While details of the negotiations remain unclear, it is possible they are linked to a £400 million debt dating back to the 1970s owned to Iran by the UK
The Government accepts it should pay the “legitimate debt” for an order of 1,500 Chieftain tanks that was not fulfilled after the shah was deposed and replace by a revolutionary regime.
Foreign Secretary Liz Truss told Sky News on Wednesday that it is a “priority to pay the debt that we owe to Iran”.
The Tehran regime remains under strict sanctions, however, which has complicated efforts to repay the money.