Siamak Namazi, an international relations expert and business consultant, was arrested as he arrived in Tehran a year ago.
His father, former UN Children's Fund employee Baquer Namazi, was detained in February when he came to Iran to seek his son's release.
Video footage of Namazi's arrest was published by Mizan, which is linked to the judiciary, earlier this week.
It was presented with dramatic music alongside images of his US passport and shots of Washington Post journalist Jason Rezaian who spent two years in prison on espionage charges.
State television reported Zaka's arrest in November 2015 on suspicion of having "numerous deep links to the US military intelligence community".
Baquer Namazi's former employer UNICEF issued a statement last month saying it was "deeply concerned about his health and well-being".
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