The New York Times has quoted White's mother, Joanne, as saying she learned three weeks ago that her son was being held at an Iranian prison.
Iran has not yet made public the arrest of White and no specific charges have been made clear.
Iran's Foreign Ministry confirmed on Wednesday that an American citizen had been arrested in the city of Mashhad, but denied reports he had been badly treated, according to the Islamic Republic News Agency.
According to his family, White had traveled to Iran three times to visit an Iranian woman believed to be his girlfriend.
Iran has confirmed it is holding a US Navy veteran - a development likely to worsen relations between the two countries even further.
He was supposed to return the United States through Dubai on July 27, but he never got on the flight, she said.
Ghasemi said reports in the United States media that White was being held in poor conditions were "false and inaccurate". "Due to privacy considerations, we have no additional information to provide at this time", a State Department official told the Times via email.More news: Wawrinka comments on Roger Federer, Rafa Nadal, Novak Djokovic's dominance
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"I'm very anxious about his health", White's mother told CBS on Tuesday.
It was based on the account of a former Iranian prisoner who said he met Mr White at Vakilabad Prison in Mashhad in October. "We have no higher priority than the safety and security of USA citizens overseas".
The network reported that it learned that Iran suggested a possible prisoner exchange past year, but State Department officials will not say whether the Trump administration is willing to negotiate.
At least three more USA citizens are being held in Iran, in addition to White.
In July 2017, Iran said that it had detained Chinese-American, Xiyue Wang, a Princeton graduate student who was conducting research in Iran.
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. "I couldn't talk about him in Iran because I am a former political prisoner and that could put my life in danger".
Those releases came months after the global deal to restrict Iran's nuclear program - the same deal from which the United States withdrew past year.