Mr Qassemi said Iran had informed the U.S. government about Mr White's arrest within days of when he was taken into custody and denied that he had been poorly treated.
White's family said he suffers from acute asthma and shortly before his latest visit to Iran he underwent chemotherapy for a neck tumour.
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.
White's mother, Joanne, told the New York Times her son had visited Iran "five or six times" to meet his Iranian girlfriend.
White's imprisonment could further worsen relations between Washington and Tehran, longtime foes.
Ghasemi denied allegations that White, who hails from Imperial Beach, California, has been mistreated in prison.
"I'm very anxious about his health", White's mother told CBS on Tuesday.
"Due to privacy considerations, we have no additional information to provide at this time", the representative said.More news: Lucky Lincoln: Rare 1943 Penny Could Fetch Over $1 Million at Auction
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Michael White's wife, Lupe White, said she last saw her husband July 9, and he texted her a day later saying he was on a business trip for two weeks.
She said the State Department had not contacted her and she was unaware he was in prison until reached by CNN.
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".
Baquer Namazi, a retired UNICEF official, and his son Siamak, an Iranian-American businessman, were sentenced in 2016 to 10 years in prison for spying and cooperating with the USA government.
In 2007, a former Federal Bureau of Investigation agent, Robert Levinson, went missing in Iran.
The prisoner issue has long added to the tensions between Iran and the United States.
Relations between Iran and the USA deteriorated sharply past year when President Donald Trump withdrew from an worldwide accord on Iran's nuclear programme and re-imposed sweeping sanctions.
President Trump's hardline stance - exiting the nuclear deal and sanctioning Iran - has not stopped it from continuing to detain three other Americans, including Siamak Namazi and his 82-year-old father.