Nick Clegg has not ruled out backing an independent inquiry into allegations surrounding Jimmy Savile.
The Deputy Prime Minister told MPs he believed it was important police investigations were allowed to finish before any further probe was set up.
But under questioning by Labour deputy leader Harriet Harman, Mr Clegg said there "may be a case for an inquiry".
Ms Harman repeated Labour calls for a broad inquiry into all aspects of the allegations made against Savile in recent days.
Mr Clegg told MPs: "I certainly accept there may be a case for an inquiry and if an inquiry were to be held which is as broad ranging as you suggest it should be, it should be independent to look at the full range of the shocking revelations as they have come to light.
"So we are not ruling it out, I just think the first priority must be to allow the police to conduct its work into what are deeply shocking and troubling revelations and allegations."
Mr Clegg continued: "I constantly ask myself how on earth is this possible on such a scale over such a long period of time in so many different settings.
"It is in many ways the sort of thing, the dark side of the cult of celebrity which maybe intimidated people from speaking our earlier. Now we know these things and they are coming to light we should proceed in a way which is led by what the police find and keep an open mind on the issue of an inquiry."
Ms Harman had asked: "Because it has come to light (Savile) carried out these crimes at the BBC but also at other institutions do you agree we need one inquiry that looks into what happened in each of these institutions to see if there were patterns of systemic failure so we get a coherent picture?
"Do you agree any inquiry must be completely independent, that is the very least the victims of Savile would expect to get to the truth and learn the lessons?
"Will the Government now set up an independent inquiry?"
Replying to Mr Clegg, Ms Harman followed up: "Can I put it to you that because the police are carrying out very important investigations which obviously should not be impeded, it does not mean that an independent inquiry shouldn't be set up now.
"I would ask you to reflect on that because daily revelations are coming forward and the victims of these abuses need to hear firmly the truth is going to be discovered.
"Can I assure you, we would be ready to discuss terms of reference to make sure we have the full and thorough inquiry which is no less than what the victims deserve."
Mr Clegg said: "You said, reasonably enough, there is no reason you can't establish an inquiry while the police are doing their work.
"But I think the issue actually is slightly the other way around - what kind of work can the inquiry do while the police are conducting their investigations.
"I don't think we should imagine that an inquiry which cannot pursue certain avenues of inquiry because the police have their own investigations would necessarily be the best answer for the victims at this time.
"But let us at least agree, firstly we must do everything we can to make sure proper answers are given to the victims and secondly I'm grateful for your signal you are prepared to work on a cross party basis on this as we work together to get to the bottom of what on earth happened."