Culture Secretary Maria Miller has warned that the Jimmy Savile sex abuse scandal raises "very real concerns" about public trust in the BBC.
In a letter to BBC Trust chairman Lord Patten, Ms Miller said it was vital that two independent inquiries were "able to follow the evidence wherever it takes them".
In his response, Lord Patten wrote: "You know how seriously the Trust takes the allegations surrounding Jimmy Savile and the need to maintain public trust in the BBC."
The peer said the inquiries would be "comprehensive and independent", but also delivered a thinly veiled warning that the Government should not wade into the row. "I know that you will not want to give any impression that you are questioning the independence of the BBC," he wrote.
The exchange came after the BBC revealed it is investigating nine allegations of "sexual harassment, assault or inappropriate conduct" among current staff and contributors as the fallout from the Savile scandal continues.
The disclosure came after director general George Entwistle was urged to "get a grip" on his organisation during a hostile grilling by MPs about the broadcaster's handling of claims of sexual abuse by the presenter over several decades.
Conservative MP Sir Roger Gale, a former producer and director of current affairs programmes for the BBC, suggested Lord Patten and Mr Entwistle may have to "fall on their swords".
He said: "Chris Patten is an old friend and a former parliamentary colleague for whom I have had a high regard but in his comment he has made it clear that he is 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. The 'Auntie knows best' line simply does not wash any more."
Sir Christopher Bland, a former BBC chairman, said: "What it needs to do is await the result of the two inquiries because at the moment the noise of people jumping to conclusions is almost overwhelming and the truth is we don't know the truth. It needs to be absolutely forensic and ruthless in its search for the truth." 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."