The broadcaster said the 64-year-old Gambaccini, who has worked at the BBC for four decades and is one of Britain's best-known music presenters, was questioned on Tuesday.
The BBC said he was released on bail later that day.
Police didn't confirm the report, saying only that two men, aged 64 and 74, were arrested as part of an operation probing historical sexual abuse.
The investigation was triggered by the abuse scandal surrounding late BBC broadcaster Jimmy Savile.
Gambaccini denied the claims on Friday, saying in a statement: "On Monday night, 28 October, I attended an excellent production of the Kander and Ebb musical, the Scottsboro Boys, at the Young Vic theatre.
"It concerned a group of black men in Alabama in the 1930s who were falsely accused of sexual offences. Within hours, I was arrested by Operation Yewtree. Nothing had changed, except this time there was no music."
A spokesman for Gambaccini said: "Mr Gambaccini was interviewed by Operation Yewtree officers about historic allegations. He answered their questions and was co-operative. He denied all allegations."
The BBC said Gambaccini wouldn't be presenting his radio show on Saturday.Do you have any story leads, photos or videos?