Richard Ratcliffe, the husband of the British-Iranian charity worker, said the family had not been updated but welcomed the signals from Tehran over the long-running dispute as "a good sign".
The Foreign Office said "legal discussions are ongoing" over the debt despite the claim made on Iranian state TV, which cited an anonymous official.
It was said that the UK government's position had not changed over the weekend and that Iran had made the claim before without the mother of one being released.
Mr Ratcliffe, who has campaigned for the release of his wife after her detention in 2016, told the PA news agency: "We haven't heard anything.
"It's probably a good sign that it's being signalled, just as last week's sentence was a bad sign.
"But it feels part of the negotiations rather than the end of them."
Earlier in the day, Foreign Secretary Dominic Raab said the debt "is not actually the thing that is holding us up at the moment".
The dispute dates back to the 1970s when the then-shah of Iran paid the UK £400 million for 1,500 Chieftain tanks.
Britain refused to deliver the tanks to the new Islamic Republic when the shah was toppled in 1979, but kept the cash despite British courts accepting it should be repaid.
Hopes were raised when Iranian state TV reported that the UK had agreed to pay the debt to see the release of the 42-year-old.
The anonymous official was also quoted saying a deal had been made between the US and Tehran for a prisoner swap in exchange for the release of $7bn (£5bn) of frozen Iranian funds.
But Washington denied the report, saying suggestions of a prisoner swap were "not true".
A Foreign Office spokesman said: "We continue to explore options to resolve this 40-year-old case and will not comment further as legal discussions are ongoing."
Mrs Zaghari-Ratcliffe, of north London, was detained in Tehran in 2016 while taking daughter Gabriella to see her family, as authorities made widely refuted allegations of spying.
She completed a five-year sentence in March, having carried out hunger strikes in protest over her treatment in jail as diplomatic efforts were made to secure her freedom.
But she and her family were delivered a fresh blow last week when she was sentenced to another year in prison and banned from leaving Iran for a further year.
Labour MP Tulip Siddiq, who represents Mrs Zaghari-Ratcliffe's constituency of Hampstead and Kilburn, told Sky News: "We weren't aware at all of any kind of deal that was meant to be taking place.
"I was caught off guard when I read the story for the first time online as was Nazanin's family.
"What I do know is that the ambassador in Tehran went to visit Nazanin this morning at her house, but they didn't mention anything about the deal.
"The discussions were about COVID and whether Nazanin would receive the vaccine. There was no mention of any negotiations and nothing about paying the debt.
"However, what I would say is that Nazanin's family are not letting their hopes get too high."
Analysis: Diplomatic editor Dominic Waghorn
Iran claims 'most likely a negotiating tactic'
The best that can be said for this development is it shows negotiations over Nazanin's fate are still going on.
It's most likely a negotiating tactic by the Iranians to put pressure on the UK. The prospect of Nazanin being released they hope will chivvy British negotiators along as they continue discussions with Iran.
The Iranians have made it clear Nazanin's fate is bound up with the debt being settled. Today it has done so with unmistakable clarity.
Clearly both sides are talking about the two issues but it seems no breakthrough yet.
Without one, Nazanin faces an agonising two more years before any chance of being reunited with her husband Richard and their daughter Gabriella.