Allegations that Savile, who died a year ago, sexually abused young girls have emerged in the past week.
It is not known when or what the name change would be, but any change would have to be agreed by their boards of trustees. The charities are also said to be considering contributing funds towards other charities that support victims of sexual abuse.
Both organisations set up in his name, pledge to continue Savile’s charitable work, which included fundraising and volunteering.
The organisations’s websites, which are simple holding pages, state: “Jimmy’s charitable trust will live on as he wished.”
Dr Roger Bodley, a trustee of the Jimmy Savile Charitable Trust and Jimmy Savile Stoke Mandeville Hospital Charitable Trust, told the Sunday TImes: “We will be meeting soon to discuss how to best use the funds that remain at our disposal. We are actively looking at supporting, among others, charities that work with survivors of sexual abuse. If there were transgressions … then we have to make amends and atone for it.”