The reporter recounted a conversation with the monarch where she said she had previously raised the issue with a senior Cabinet minister.
A BBC spokesman said: “Our correspondent Frank Gardner revealed details of a private conversation, which took place some years ago with the Queen. It was wholly inappropriate [to do so]. Frank is extremely sorry for the embarrassment caused and has apologised to the palace.”
Gardner, the BBC’s security correspondent, had told the radio programme: “This is a conversation we had a little while ago and she did say that she had mentioned to – I don’t know which home secretary it was at the time – that was there not some law he had broken?
“I wouldn’t say she was necessarily lobbying – that’s not for me to say – but, like anybody, she was upset that her country and her subjects had been denigrated by this man who was using this country as a platform for his very violent, hateful views.”
A palace spokesman declined to comment.