David Cameron has raised the prospect of Sir Jimmy Savile being posthumously stripped of his knighthood in the wake of allegations of sexual abuse against young girls.
The Prime Minister stopped short of revealing whether he felt the former DJ should lose the honour, but suggested that the case should be considered by a Whitehall committee which has the power to recommend forfeiture.
His comments came after the chairman of the BBC Trust gave his backing to inquiries by police and the corporation.
Lord Patten said the allegations against Savile could not be excused as behaviour from a time when "attitudes were different".
He told a business dinner in Cardiff that it was "no excuse to say 'That was then' in the 1960s, '70s and '80s, and attitudes were different then.
"It's no excuse to say 'I'm sure the same thing used to happen with pop groups and others at the time'. Those things may be true but they don't provide an excuse."
The radio and TV presenter and charity fundraiser, who died in 2011, has been accused by a growing number of women in the last few weeks of sexual abuse over a number of years.
Mr Cameron told ITV1's Daybreak programme: "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, how they were allowed to happen, and then of course everyone has to take their responsibilities."
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."
BBC Director-General George Entwistle apologised to victims of Savile's alleged sex abuse and pledged the corporation would hold its own inquiry following a police probe.
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