David Cameron raised the prospect today 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 last night 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 today: "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."
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 yesterday apologised to victims of Savile's alleged sex abuse and pledged the corporation would hold its own inquiry following a police probe.
Lord Patten echoed Mr Entwistle's announcement and outlined how the BBC would act.
He said: "Immediately these allegations came out, we went to the police, we agreed with Acpo (the Association of Chief Police Officers) the way we should handle this and they told us the police inquiry should come first and we shouldn't undertake our own investigation until there had been a police inquiry or we might get in the way of it.
"So there will be a full police inquiry and we will encourage people to co-operate with it and, when that is completed, we will then look at the issues which still remain to be resolved in a way which will have to command credibility in the wider community.
"Because the BBC exists above all on trust and the relationship between the wider public and the BBC itself. And when the BBC is at its best, it's not only because it is providing terrific, creative, challenging TV and radio, it is because the public think they own it and can identify with it. Just think of the Olympics, the torch, the cultural Olympiad - all of those helped bring the community together."
Mr Entwistle, who started in his new role last month, spoke of deep regret about the ordeals of the women involved in the "awful allegations", and said there would be a "comprehensive examination" of what went on.
He spoke out a day after Mr Cameron called for the "truly shocking" allegations against Savile to be fully investigated.
Last week the BBC said it would work with police in examining the claims and on Friday Mr Entwistle wrote to staff urging them to come forward with information.
Speaking on the BBC Radio 4 Today programme yesterday, he said: "These are awful allegations that have been made, and they are criminal allegations.
"And the first thing I want to say is that the women involved here have gone through something awful, something I deeply regret they should have to go through, and I would like to apologise on behalf of the organisation to each and every one of them for what they've had to endure here."
Mr Entwistle said any BBC inquiry would take place after police had carried out their investigations.
"When the police have finished everything they have to do, and when they give me an assurance that there is no danger of us in any way compromising or contaminating an investigation, I will take it further and ensure that any outstanding questions are answered properly," he said.
Any BBC probe, he added, would examine the "broad question of what was going and whether anybody around Jimmy Savile knew what was going on".
A growing number of women have come forward to claim they were either abused - many of them under-age - or that they saw others who were victims. A number of former colleagues have told how they were aware of rumours about the former Top Of The Pops presenter.
Unease about the claims has led to a number of memorials to the star being removed. Yesterday it was announced that an inscription on the wall at Leeds Civic Hall in recognition of his charity work will be taken down.
A street sign in Scarborough, North Yorkshire, has also been taken away.
Mr Entwistle said Savile was widely regarded as a "bit peculiar", but he said if anyone had been directly aware of the allegations they should have spoken up.
Mr Entwistle said: "It's very important that people don't think the BBC of today is anything like in character managed the way it was at the time."
Former Radio 1 DJ Liz Kershaw said at the weekend that when she joined Radio 1 in 1987 - the year Savile left - his behaviour was an "open secret" at the station.
She described how she was routinely groped by another presenter as she was broadcasting.
Broadcaster Janet Street-Porter previously revealed that she was aware of rumours about the television and radio presenter's alleged abuse of under-age girls when she worked at the BBC in the late 1980s.
Police child abuse officers have met BBC officials to discuss the allegations concerning Savile.
Officers said they were contacting all individuals who have made claims about the late presenter and should know how many reported victims there are some time this week.
Video: When Louis Met Jimmy Savile