BBC Trust chairman Lord Patten has sought to reaffirm the corporation's independence from government, in a letter to the culture secretary.
Maria Miller had written to express concerns about public confidence in the BBC over inaccurate statements about Newsnight's scrapped investigation into Jimmy Savile.
Lord Patten said the BBC's inquiries would be comprehensive and independent.
This comes after BBC director general George Entwistle appeared before MPs.
He told the Commons culture committee on Tuesday that the past "culture and practices of the BBC seemed to allow Jimmy Savile to do what he did".
The director general said the Newsnight probe into sexual abuse claims against former BBC presenter and DJ Savile, who died last year aged 84, should not have been dropped.
But he said he did not believe management pressure had led to the report being shelved last December.
Police have launched a criminal investigation into Savile. They have described him as a predatory sex offender and believe he may have abused many people - including young girls - over a 40-year period.
Since the claims about Savile came to light, charities say they have seen an increase in the number of people calling their helplines about sexual abuse inflicted many years ago.
'Out of touch'
Mrs Miller told MPs last week that the BBC was taking the sexual abuse allegations against Savile very seriously and welcomed its decision to set up three independent inquiries.
But in a letter to Lord Patten on Tuesday, she said "very real concerns" were being raised about public trust in the BBC.
She said this followed revelations of factual inaccuracies in the original explanation for Newsnight dropping its Savile investigation.
She added: "Both the trust and the executive must stand ready to act on the results of the inquiries swiftly and decisively.
"It is the trust's role as the sovereign body of the BBC to be accountable to licence fee payers and ensure public confidence in the BBC."
In response, Lord Patten said: "You have recognised both the credibility and the scope of those who are leading the inquiries and the wide scope of their terms of reference."
But he warned: "I know that you will not want to give any impression that you are questioning the independence of the BBC."
Conservative MP Sir Roger Gale, a former BBC producer and director, said Lord Patten's response showed he was "out of touch, not only with the strength of feeling and concern in Parliament about the 'Savile affair' and related matters but, more importantly, with the strength of public revulsion at what has happened at Television Centre and with the corporate culture that - for the best part of 40 years - has apparently covered it up".
"Attack may be the best form of defence but in seeking to criticise a culture secretary who has not, ever, sought to challenge the independence of the BBC, he indicates how very little, within that corporate arrogance, has really changed," he added.
Former BBC chairman Sir Christopher Bland stressed the need to wait for the inquiries' results "because at the moment the noise of people jumping to conclusions is almost overwhelming and the truth is we don't know the truth".
But he added: "If you followed up every rumour or allegation about the BBC as chairman or director general, you would have no time for your proper job."
BBC political correspondent Norman Smith said there would be pressure on the inquiry into the BBC's management of the Newsnight investigation to release at least interim findings within days rather than weeks.
"At Westminster I think you will find a broad swathe of opinion, from people who are pro-BBC to people who are much more critical of the BBC, who just think this needs to be cleared up an awful lot quicker than the current timetable the BBC is working to," our correspondent said.
Mr Entwistle told MPs he had learned about the Savile investigation by Newsnight while he was head of BBC Vision - responsible for TV scheduling and commissioning - but did not press for more details as he did not want to show "undue interest" in a news issue.
He said Newsnight editor Peter Rippon had decided to drop the report on Savile "on his own account" and there had been "no external pressure".
In his blog earlier this month defending his decision to pull the report, Mr Rippon said he had been "guided by editorial considerations only", adding some of the team "disagreed strongly with [his] judgement" while others "agreed equally strongly".
On Monday the BBC issued a correction to some specific elements of the blog, calling it "inaccurate or incomplete in some respects".
Mr Entwistle said he had asked Mr Rippon to step aside because of those inaccuracies.
Meanwhile, a woman has told BBC Radio 5 live's Tony Livesey programme she was raped by Savile in his caravan in a car park on BBC premises in 1970 when she was 22 and had just won a beauty contest.
She said on another occasion she was contacted by the same BBC employee who had put her in touch with Savile and invited to attend a film - and then that man raped her in a BBC office.
The woman, who wanted to remain anonymous, said she did not speak out at the time because she thought nobody would have believed her.
She told the BBC and the police about the alleged attacks this month.
The BBC has launched two inquiries related to Savile - one into whether there were any failings in the BBC's management of the Newsnight investigation, and one into the culture and practices of the BBC during Savile's time at the corporation.
In a third review, an expert will look at sexual harassment claims and practices at the corporation.
Conservative MP Ann Main has tabled a Commons motion calling for the Leveson Inquiry into press phone hacking to be extended to scrutinise the BBC.
The motion argues "the BBC should not dictate the terms of an inquiry into itself".