Nazanin Zaghari-Ratcliffe, who has been held in an Iranian jail for more than 18 months, is eligible for early release.
Mrs Zaghari-Ratcliffe told her husband Richard that she had been informed of the news by her Iranian lawyer.
Her case has been marked on the Iranian judiciary database as “eligible for early release”.
Mr Ratcliffe told Sky News: “This is definitely a positive step. I am a bit cautions but this is terrific news. There has not been confirmation yet and there is still some paperwork to complete.
“But compared to where we have been in recent weeks, there is absolutely no doubt that this is positive news, fingers crossed. My job is not to get too excited but it all does seem much more positive.”
Mr Ratcliffe said that once his wife was home there would have to be “lessons learned but that’s for another day”.
Mrs Zaghari-Ratcliffe, who is British-Iranian, was arrested at the airport in Tehran after visiting her family with her daughter in 2016.
She is serving five years in prison after being charged with planning to topple the Iranian government, which she denies.
Boris Johnson, the Foreign Secretary, met with Iranian Foreign Minister Mohammed Javad Zarif in Tehran earlier in December to discuss Mrs Zaghari-Ratcliffe’s case.
Before his intervention, Mr Johnson was forced to apologise after he said he had understood she was “teaching journalists” while in Iran, which she denies.
Authorities used this as justification for new charges against her of “spreading propaganda”.
Mrs Zaghari-Ratcliffe’s husband said she could face another 16 years in prison.
The family’s hopes were heightened when she was due to go on trial at the beginning of December but this did not go ahead.
At the time her husband Richard said it offered a “ripple of freedom”.