The premier's intervention came as the editor of the internationally renowned media organisation's flagship "Newsnight" show stepped aside after admitting he had given an "inaccurate" account of why the BBC had axed its own expose of the alleged abuse of underage girls by the late Jimmy Savile.
"These are serious questions. They need to be answered," Cameron said of the cover-up allegations.
The scandal has engulfed the BBC at a time when it remains under pressure from its critics - which include much of the conservative media - who have queried whether it should still be funded via an annual licence fee paid by the public.
Critics, most notably media magnate Rupert Murdoch's son James, have said the licence fee gives the BBC an unfair edge over private competitors. The BBC is already cutting its workforce and output after Cameron's government imposed deep spending cuts and any loss of public trust could prove an issue in future discussions over funding and the licence fee.
While Savile, who died last year, was little known beyond Britain, the eccentric, cigar-chomping one-time DJ was one of the most recognised TV personalities on British television in the 1970s and 80s, hosting prime-time children's and pop shows.
But Savile, who was knighted by Queen Elizabeth for his charity work and was famous for his garish outfits and long blonde hair, is now accused of raping and abusing girls as young as 12, some on BBC premises at the height of his fame.
Critics argue that the BBC covered up his alleged crimes which police say took place over six decades and were on an "unprecedented scale".
"The developments today are concerning because the BBC has effectively changed its story about why it dropped the Newsnight programme about Jimmy Savile," said Cameron when asked about the issue following a speech in London.
Veteran BBC foreign correspondent John Simpson said the organisation's handling of the case was the worst crisis to hit the corporation in his almost 50-year career.
"I don't think the BBC has handled it terribly well," he told the "Panorama" programme in clips released by the BBC.
"All we have as an organisation is the trust of people, the people that watch us and listen to us and if we don't have that, if we start to lose that, that's very dangerous."
The imbroglio has piled pressure on new BBC chief George Entwistle to explain what happened.
Entwistle only replaced Mark Thompson, the incoming New York Times Co's chief executive, as BBC Director General in August. He will appear before lawmakers on Tuesday amid a growing media clamour for answers.
"INACCURATE OR INCOMPLETE"
The claims about Savile were first aired on rival channel ITV at the end of September but the BBC faced embarrassment when it later emerged that its own programme - "Newsnight" - had carried out an investigation into Savile last year but had never broadcast its findings.
Newsnight editor Peter Rippon, who stood aside on Monday, wrote in his blog that the decision to drop the programme was made for editorial reasons, and that the investigation had only focused on possible institutional failings by police and prosecutors.
But those reasons have been publicly disputed by the show's journalists and Rippon stood down to allow an independent inquiry to try to establish the truth.
The BBC conceded Rippon's explanation was flawed. "The explanation by the editor in his blog of his decision to drop the programme's investigation is inaccurate or incomplete in some respects," it said in a statement.
The inaccuracies referred to have fuelled suspicions that BBC chiefs pulled the Newsnight probe because they did not want it to clash with planned programmes over Christmas commemorating Savile's life as a TV celebrity and charity fundraiser.
Another BBC documentary show "Panorama", due to air in Britain on Monday night, is expected to give details of how much information the Newsnight team had on Savile at the time their investigation was shelved.
Meirion Jones, the producer behind the Newsnight story, told Panorama he had warned his editor that the BBC was at risk of being accused of a cover-up if it did not run the story.
"We weren't asked to find more evidence, or anything like that, we weren't asked to get more people on camera, we were told to stop working on the story," he said in clips broadcast by the BBC.
The BBC has launched two independent reviews of the allegations, one looking into Savile's actions and another to investigate why the Newsnight report was dropped.
It has not commented officially on the case, saying it would be inappropriate to say anything until the reviews have been concluded.
"I'm afraid I can't make any comment on Panorama because I haven't seen it yet," BBC boss Entwistle told reporters when he left his home for work. "I will of course be taking questions at (parliament's) culture select committee tomorrow."
Media consultant Steve Hewlett, a former "Panorama" editor himself, said he did not believe there had been a great conspiracy but said the BBC had got itself in a mess.
"What the BBC have said about it is partial to the point of being misleading which has the effect of fuelling concern about what really went on," he told Reuters.
"They are now dealing with the way they've handled it and what they've said about it, rather than the fundamentals of what happened."
