Scotsman Andrew Stimpson, 25 was diagnosed HIV-positive in 2002 but was found to be negative in October 2003.
Mr Stimpson, from London, said he was "one of the luckiest people alive".
Chelsea and Westminster Healthcare NHS Trust confirmed the tests were accurate but were unable to confirm Mr Stimpson's cure because he had declined to undergo further tests.
A statement from the trust said: "This is a rare and complex case. When we became aware of Mr Stimpson's HIV negative test results we offered him further tests to help us investigate and find an explanation for the different results.
I think I'm one of the luckiest people alive
Bacteria modified to fight HIV
"So far Mr Stimpson has declined this offer."
A trust spokeswoman added: "We urge him, for the sake of himself and the HIV community, to come in and get tested.
"If he doesn't feel that he can come to Chelsea and Westminster then he should please go to another HIV specialist."
There have been anecdotal accounts before from Africa of people shaking off the HIV virus.
Mr Stimpson, who is originally from Largs in Ayrshire, said: "There are 34.9 million people with HIV globally and I am just one person who managed to control it, to survive from it and to get rid of it from my body.
"For me that is unbelievable - it is a miracle. I think I'm one of the luckiest people alive."
Mr Stimpson told the News of the World and Mail on Sunday that he became depressed and suicidal after being told he was HIV-positive but remained well and did not require medication.
Some 14 months later he was offered another test by doctors, which came back negative.
He sought compensation but has apparently been told there is no case to answer because there was no fault with the testing procedure.
He has told the papers he would do anything he could to help find a cure.
Deborah Jack, chief executive of the National Aids Trust, said: "This appears to be a highly unusual case and without further tests it is impossible to draw any conclusions for people living with HIV.
"The virus is extremely complex and there are many unknowns about how it operates and how people's bodies react to it.
"Therefore, if this case were able to shed further light, it could be extremely valuable for research into treatments or a cure."
Aids expert Dr Patrick Dixon, from international Aids group Acet, said the case was "very, very unusual".
"I've come across many anecdotal reports of this kind of thing happening in Africa, some quite recently, but it's difficult to verify them," he told BBC News 24.
"You have to be rock-solid sure that both samples came from the same person, no mix-up in the laboratory, no mistakes in the testing, etc.
"This is the first well-documented case."
He said the case was important because "inside his immune system is perhaps a key that could allow us to develop some kind of vaccine".
The article mentions that there have been people in Africa which dodged the HIV virus. In Microbiology class in high school, we learned this was because they lacked the receptor sites for the HIV virus so the virus could never attach itself, therefore they couldn't get it. Perhaps the man's first HIV test was taken when it was still in the blood stream; second taken when it was no longer present in his system.
Don't be so simplistic MWpoo ... I'm harmless. If we're gonna make an impact with the removal of a single family then pick one with a high degree of self serving destructive tendencies like the Bush dynasty. Then at least you would leave a positive result.