Domenico Rancadore is wanted in Italy to serve a seven-year jail term for his role in the Sicilian mafia.
Italian authorities are seeking his extradition on charges including extortion, racketeering and drug trafficking at a two-day hearing at London's Westminster Magistrates Court.
The 64-year-old, dressed in a grey and blue cardigan, told court he moved to Britain to give his "children a good life", claiming "their life wasn't secure" in Italy.
Known in the mafia as "The Professor", Rancadore had been on Rome's list of most dangerous criminals.
He had been living in a modest suburban house in west London under the name of Marc Skinner, with his English wife and two children, when he was arrested on August 7th last year under a European arrest warrant.
Rancadore told the court: "I changed my name, I changed my life, I didn't want to go back to Italy."
Anne Skinner, Rancadore's wife, said her husband's health was deteriorating. He had a stent fitted for a heart problem in 2012.
She said he now suffers memory loss and that his hands tremble "a lot".
"If my husband were to go to Italy it would be difficult to visit him. It would be devastating for me," she added.
Prosecutors told an earlier court hearing that Rancadore was a leader in a gang that "spread terror" on the Italian island of Sicily, "systematically murdering anybody who did not comply with the will of the members of the organization".
Rancadore was convicted in his absence of Mafia links in Italy after he went on the run and moved to Britain in 1993.
He was granted conditional bail last November, but the decision was overturned in the High Court.
Judge John Goldring said there was "a real risk -- indeed a very real risk -- that he would abscond if granted bail".