The Top Gear presenter told The One Show yesterday that public sector workers going out on strike on November 30 should be "executed in front of their families".
Trade union UNISON later called on the BBC to sack Clarkson and suggested that it may contact the police regarding the remarks.
Prime Minister David Cameron, known to be a friend of Clarkson's, dismissed the broadcaster's remarks as "a silly thing to say".
"I didn't for a moment intend these remarks to be taken seriously - as I believe is clear if they're seen in context," Clarkson has now said.
"If the BBC and I have caused any offence, I'm quite happy to apologise for it alongside them."
The BBC also said: "As has now been widely reported, we had many complaints about a number of Jeremy Clarkson's comments on the show.
"The One Show is a live topical programme which often reflects the day's talking points. Usually we get it right, but on this occasion we feel the item wasn't perfectly judged.
"The presenters apologised at the end of the programme to viewers who were offended by his comments and the BBC and Jeremy would like to apologise for any offence caused."