Security correspondent Frank Gardner told BBC Radio 4 that the Queen had raised concerns with ministers about cleric Abu Hamza al-Masri, who the European Court of Human Rights yesterday (September 24) said could be extradited to the US.
Gardner told the Today programme that the Queen had discussed Abu Hamza with a previous Labour home secretary and that she was upset the cleric had not been arrested.
"The Queen was pretty upset that there was no way to arrest him," he said. "She couldn't understand - surely there must be some law that he broke.
"Well, sure enough there was. He was eventually convicted and sentenced to seven years for soliciting murder and racial hatred."
He added that the Queen was not lobbying the government with her comments.
A letter from the broadcaster to Buckingham Palace said that both the BBC and Gardner regretted the breach of confidence and that the revelations were "wholly inappropriate".
It added that Gardner had apologised for the revelations and that the conversation should have been kept private.
Buckingham Palace and the Home Office have declined to comment on the matter.