White is the fourth American being held by Iran, and while CBS News has learned that Iran floated the idea of a prisoner exchange past year, State Department officials will not say whether the Trump administration is willing to negotiate.
Iran has confirmed it is holding a US Navy veteran - a development likely to worsen relations between the two countries even further.
The US State Department said Tuesday it was "aware of reports" of his detention, but declined to release details, citing privacy considerations.
According to his family, White had traveled to Iran three times to visit an Iranian woman believed to be his girlfriend.
On January 9, Iranian state news agency IRNA carried a statement by Foreign Ministry spokesman Bahram Qasemi that confirmed the arrest, but did not specify when it had happened or what crime he was accused of.
"I'm very anxious about his health", White's mother told CBS on Tuesday.
Among them is Xiyue Wang, a Princeton University student, who was given a 10-year sentence for espionage. She called them when her husband did not return home, she said.
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JLR has warned of further losses in the event of the United Kingdom leaving the European Union with a poor trade deal or no deal. Governor of the Bank of England Mark Carney said the probability of a no-deal Brexit is "uncomfortably high".
Mr Qassemi said Iran had informed the United States government about Mr White's arrest within days of when he was taken into custody and denied that he had been poorly treated.
His family say he suffers from acute asthma and shortly before his latest visit he underwent chemotherapy for a neck tumour.
London-based news website IranWire said Mr White was being held in Vakilabad prison in Mashhad, northeastern Iran. While the Trump administration offered to discuss prisoners previous year, a senior administrative official said Iran turned that offer down, leaving the fate of the Americans unclear.
At least three more US citizens are being held in Iran, in addition to White.
In January 2016, Iran freed four USA prisoners - Washington Post journalist Jason Rezaian, Marine veteran Amir Hekmati, Christian pastor Saeed Abedini and Nosratollah Khosravi-Roodsari - in exchange for the U.S. granting clemency to seven Iranians indicted or imprisoned in the United States.
In 2007, a former Federal Bureau of Investigation agent, Robert Levinson, went missing in Iran.
The last time Iran released any imprisoned Americans was when the global accord on Iran's nuclear programme took effect in January 2016.
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