Mr Gani has said claims he backs IS are defamatory and must be retracted.
After winning the London Mayoral election, Mr Khan, the first Muslim to hold the post, accused the Conservatives of running a “divisive, dog-whistle campaign”.
‘Unaware of clarification’
Mr Fallon made the claim about Mr Gani during a BBC interview, as he defended the Conservatives’ London mayoral election campaign.
In a tweet after the interview was broadcast, Mr Gani indicated he would be seeking legal action against the defence secretary.
When asked about the legal letter issued by Mr Gani’s lawyers, a spokesman for Mr Fallon said: “Michael made clear he was quoting a claim by Andrew Neil on a BBC Mayoral debate programme.
“He was unaware of the clarification and apology that the BBC had issued on Neil’s claim.
“Had he been aware, he would not of course have quoted him and as soon he became aware he put the record straight. He naturally apologises for this inadvertent error.”
Prime Minister David Cameron has faced calls of his own to apologise for criticising Mr Gani in Parliament.
At Prime Minister’s Questions in April, Mr Cameron said Mr Khan had appeared on a platform nine times with Mr Gani, a preacher whom the prime minister alleged “supports IS”.
Mr Gani said at the time that he wanted to take legal action against Mr Cameron but was unable to do so because of Parliamentary privilege rules, which protect MPs’ freedom of speech in the Commons.