In a statement, the BBC said it had asked its investigations unit to contact police and "offer full support" to any investigations.
The claims are made in a documentary due to be aired on ITV on Wednesday.
Earlier, Surrey Police referred an allegation of rape against the late presenter to the Metropolitan Police.
"A woman yesterday (Monday, 1 October) reported a historic allegation of rape," the Surrey force said in a statement.
"The alleged offence occurred in London, therefore the matter has been referred to the Metropolitan Police."
The Met has issued a statement which confirmed the allegation, believed to date back to the 1970s, "is currently under review by the Met police's Sapphire unit - which investigates rape and sexual offences".
The BBC statement said: "We are horrified by allegations that anything of this sort could have happened at the BBC - or have been carried out by anyone working for the BBC."
"They are allegations of a serious criminal nature which the police have the proper powers to investigate."
It added that police forces in receipt of allegations would be offered help in investigationg the claims and full support would be provided "to any lines of inquiry they wish to pursue".
Writing in a BBC Editor's Blog, Newsnight editor Peter Rippon addressed newspaper reports that the programme had been ordered to drop an investigation into the presenter's alleged sexual abuse.
Mr Rippon said: "It has been suggested I was ordered to do it by my bosses as part of a BBC cover-up. It has also been suggested that we deliberately withheld information from the police.
"Both these allegations are totally untrue and despite consistent strong denials keep getting repeated."
Mr Rippon went on say Newsnight had investigated a claim by a key witness that the Crown Prosecution Service had dropped an investigation into sex abuse claims on account of Sir Jimmy Savile's age.
He wrote: "The CPS told us: 'The CPS reviewing lawyer advised the police that no further action should be taken due to lack of evidence.'
"The additional guidance noted stated. 'As this is the case, it would not be correct to say that his age and frailty was the reason for no further action being taken.'
"I took the decision not to publish."
Mr Rippon went on to admit "There were some of my team who disagreed strongly with my judgement, and others who agreed equally strongly.
"However, those who disagreed accepted my decision. There were no rows of any kind as has been reported."
Police also said on Tuesday that the Leeds-born DJ and TV presenter was investigated during an inquiry into abuse at a Jersey children's home.
The three-year inquiry into child abuse at Haut de la Garenne heard allegations that Sir Jimmy was involved in an indecent assault in the 1970s.
The States of Jersey Police said there was insufficient evidence for the investigation to proceed.
Sir Jimmy, who died last year aged 84, was never charged with any abuse offences.
An ITV1 documentary - Exposure: The Other Side of Jimmy Savile, to be broadcast on Wednesday - accuses Sir Jimmy of sexual abuse against under-age teenage girls.
It will show several interviews with alleged victims of the DJ and TV presenter.
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