The Jimmy Savile Charitable Trust is considering changing its name after allegations of abuse were made about the late DJ and charity fundraiser.
Several women have alleged Sir Jimmy sexually assaulted them as teenagers.
The trustees said they "have been contacted by a number of members of the public suggesting that they should change the charity's name, and they are in the process of looking into this."
Earlier Prime Minister David Cameron said the claims were "truly shocking".
In a statement the charity said: "(The trustees) are actively looking at supporting, amongst others, charities that work with survivors of sexual abuse. They feel this is the right thing to do in the circumstances."
It goes on to say: "The trustees are aware that a large proportion of the funds the charity received came from donors other than the late Jimmy Savile, through fund-raising and legacies.
Sir Jimmy died in October 2011, at the age of 84. The face of Top of the Pops in the 1960s, he hosted TV favourite, Jim'll Fix It, on BBC 1 in the 1970s and 1980s and was knighted in 1990 for his charity work.
Sexual assault claims
But in the last week allegations have emerged about serious sexual assaults upon under-age girls at the height of his fame.
Some of the allegations refer to incidents on BBC premises.
The Metropolitan Police said last week it would be assessing allegations against Sir Jimmy but had not as yet launched an investigation.
Speaking on the BBC's Marr programme on Sunday Mr Cameron said the allegations needed to be "properly looked at, properly investigated".
He added: "It seems to me it is very important that the organisation, the BBC, does that itself.
"But also, if there are questions that should be pursued by the police and other organisations, everyone has to ask themselves the question `Is there new evidence that needs to be looked at?
"Are there new things as an organisation we should look at and examine?'"
On Saturday former BBC Radio 1 DJ Liz Kershaw said she had been regularly fondled in the 1980s by another colleague, who she did not name.
And on Sunday comedian Sandi Toksvig says she was groped by a "famous individual" while she was broadcasting in the 1980s.
Ms Toksvig, 54, did not indicate where she was working at the time but said when she told staff "everybody thought it was amusing".