It comes after two hospitals spoke of their "shock" at the allegations made against the Jim'll Fix It star.
Nurses at Stoke Mandeville Hospital, in Aylesbury, Buckinghamshire, told children to stay in bed and pretend they were asleep to prevent the TV presenter from groping them, it has been claimed.
Eight criminal allegations have been made to police against Jimmy Savile
They were also said to have dreaded Savile's visits because of his behaviour.
A statement from Buckinghamshire Healthcare NHS Trust read: "We are shocked to hear of the serious allegations about Jimmy Savile.
"We are unaware of any record or reports of inappropriate behaviour of this nature during Jimmy's work with the trust."
Rebecca Owen, a former patient at the hospital, told The Daily Telegraph: "It was an air of resignation that you had to put up with.
"There was some sort of ironic chatter between the nurses about who would be the lucky one to go off to his room.
"And then, as one of the nurses was leaving or passing by my bed, she leant over and said, 'The best you can do is stay in bed until he's gone and pretend to be asleep.'"
Allegations were also made about abuse at Leeds General Infirmary relating to incidents in the 1970s.
The hospital said two people, thought to be patients, came forward to report abuse.
It said it was unclear whether the alleged abuse took place when Savile was a volunteer porter at the hospital, or when he was visiting informally.
In a separate incident, former nurse June Thornton said she saw Savile abuse a brain-damaged young girl at Leeds General Infirmary. Ms Thornton was a patient at the time of the alleged incident and was recovering from an operation.
She said: "She had brain damage, and Jimmy Savile came in and kissed her.
"He started kissing her neck, running his hands up and down her arms, and then started to molest her.
"Because I was laid flat on my back, there was nothing I could do."
A spokesman for Leeds Teaching Hospitals NHS Trust said: "We are shocked at the nature and extent of the very serious allegations made against Jimmy Savile which were revealed by the Metropolitan Police on Wednesday.