David Cameron has raised the prospect that Sir Jimmy Savile could be posthumously stripped of his knighthood amid allegations of sexual abuse against young girls.
But the move would require a change in the law as technically, when the former DJ died last year, the honour ceased to exist.
The Prime Minister stopped short of revealing whether he felt Savile should lose his knighthood, but suggested that the case should be considered by a Whitehall committee which has the power to recommend forfeiture.
"These stories are deeply, deeply troubling and I hope that every organisation that has responsibilities will have a proper investigation into what happened, and if these things did happen, and how they were allowed to happen, and then of course everyone has to take their responsibilities," Mr Cameron said.
Asked if Savile should lose his knighthood, Mr Cameron said: "We have something called a Forfeiture Committee. It is not chaired or sat on by me but it is responsible for looking at honours and the removal of honours, and obviously they have to do their job too."
But the Cabinet Office said Savile no longer had a knighthood to revoke, as individuals cease to be a member of the order when they die.
"It's a living order and then you cease to be a member when you die," a spokesman said. "There isn't an honour to revoke."
A campaign by the Sun newspaper is calling for a change in the law so that honours can be revoked posthumously by the Forfeiture Committee.
Mr Cameron said later that Savile's knighthood was "something people need to think about", but acknowledged the Forfeiture Committee would not usually consider somebody who had died.
He told ITV News: "The Forfeiture Committee exists to look at the issue of honours, but as I understand it they have never looked previously at someone who is deceased and that is the way things stand. I'm not going to take this forward right now, but this is obviously something people need to think about."