"Baquer and Siamak Namazi... have been sentenced to 10 years in prison for espionage and collaboration with the American government," said prosecutor-general Abbas Jafari Dolat-Abadi, according to the Fars news agency.
Siamak Namazi, a well-connected business consultant who has supported reformist politicians in Iran, was arrested as he arrived in Tehran a year ago.
His father, Baquer, was detained in February when he came to Iran to seek his son's release.
He is a former employee of the UN Children's Fund, or UNICEF, and served as a governor of the Iranian province of Khuzestan before the Islamic revolution of 1979.
The United States demanded the release of the Namazis.
"We are deeply concerned about reports that US citizens Siamak Namazi and Baquer Namazi have each been sentenced to 10 years in prison," State Department spokesman Mark Toner said in a statement.
"We join recent calls by international organisations and UN human rights experts for the immediate release of all US citizens unjustly detained in Iran, including Siamak and Baquer Namazi, so that they can return to their families."
The US also expressed concerns about reporters of the "declining health" of Baquer Namazi.
UNICEF voiced its "deep sadness and personal concern" over the sentence, appealing for the release of the elder Namazi on humanitarian grounds.
Fars said four others -- Farhad Abd-Saleh, Kamran Ghaderi, Nezar Zaka and Alireza Omidvar -- had also been sentenced to 10 years in jail.
- Hardliners criticise Namazis -
Zaka is the Lebanese-American arrested in November 2015 and accused on state television of "numerous deep links to the US military intelligence community".
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The hardline Kayhan newspaper has described Omidvar as an "associate" of Namazi.
Conservative media have criticised Siamak Namazi's links to the National Iranian American Council, a Washington-based NGO created by Iranian nationals aimed at boosting ties between the two countries.
At the time of his arrest last year, he was working for Crescent Petroleum, an energy firm based in the United Arab Emirates, which is involved in an ongoing legal dispute with Iran over a failed gas deal, and is seeking billions of dollars in compensation.
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.
Mizan published a long article in February, criticising the Namazi family's efforts to "influence internal events in Iran" through its financial contributions to NGOs.
It also criticised their role in promoting ties between Iranian and Western businesses during the rule of reformist president Mohammad Khatami.
Rezaian was released in January at the same time as a nuclear accord with world powers came into force, lifting sanctions on Iran.
Washington helped to airlift $400 million worth of Swiss francs and euros to Iran shortly afterwards -- part of the money owed by the US because it cancelled defence sales at the time of the revolution.
The White House has denied accusations from hardline critics that this amounted to a hostage ransom for Rezaian and four others released at the same time.
In its statement, the State Department also raised the case of Robert Levinson, a former FBI agent and ex-CIA contractor who went missing on Iran's Kish Island in 2007.
"We also respectfully underscore the importance of Iran cooperating with the United States to determine the whereabouts of Mr Robert Levinson," Toner said.
"As President (Barack) Obama stated last January, we will not rest until the Levinson family is whole again."