The Cabinet minister said he met Alexander Kashitsyn - who he now thinks was a secret agent - at the Seven Stars pub in Fulham while he had been seeking election as Tory MP for the area almost 20 years ago.
Mr Hands, who was elected MP for Chelsea and Fulham in 2010, claimed Mr Kashitsyn had first introduced himself as a Russian Embassy worker during a Conservative event in 2004 at which Boris Johnson was a guest speaker.
He said Mr Kashitsyn asked for a meeting shortly afterwards and, at the pub, probed Mr Hands for information about Iran's nuclear programme.
Speaking with Channel 4 for a Dispatches documentary, Mr Hands said: "First he'd said he wanted to talk about street-sweeping cleaning or bin collection.
"Then suddenly he asked me to get him a document about the Iranian nuclear programme from the House of Commons Library."
Mr Hands said he declined to help, adding: "I was very suspicious.
"I thought he was a Russian intelligence officer getting a soon-to-be British Member of Parliament to do something for the Russian state.
"This is sometimes a sort of a classic kind of... a sort of espionage technique.
"You get somebody to steal something or get you a document that is easy to get - not that you need the document - but the idea is to test somebody's willingness to do something for you that's essentially what the technique is."
Mr Hands said two years later, MI5 officers told him that his number had appeared on the phone of a Russian intelligence operative.
He said the officers told him there was a "real problem" with "people like Kashitsyn in London" who were "very, very active" in trying to infiltrate British politics.
Channel 4 has said the Russian Embassy declined to comment on the allegations made in their film.
Airing on Sunday at 10pm, Strippers, Spies And Russian Money: Dispatches investigates how Russia sought to influence British politics in the years before the invasion of Ukraine.