An independent review into his conduct on NHS premises published on Thursday found the ex-BBC DJ subjected patients aged between five and 75 to "truly awful" sexual abuse over decades of unrestricted access to NHS hospitals in England, including the high-security mental hospital Broadmoor, home to some of the country's most violent offenders such as Yorkshire Ripper Peter Sutcliffe.
Other shocking revelations included that Savile, who died in 2011, boasted about having sex with corpses and stealing glass eyes from dead bodies to use as jewellery.
All of the 31 health facilities named in the assurance report, compiled by Kate Lampard QC for the Department of Health, were located in England and Wales.
Some 157 reports from the police were passed to the UK's Department of Health relating to sex attacks by Savile on NHS premises. The Sunday Herald has learned, of these, three relate to Scottish hospitals. These reports were passed to NHS Scotland and then onto Police Scotland.
Police Scotland have confirmed it is carrying out "ongoing investigations", but refused to give any details of the locations or areas involved.
A spokesman for Police Scotland said: "At the present time less than 20 referrals have been received which relate to Operation Yewtree crimes having occurred within Scotland, three of which relate to hospital settings. These enquiries are subject to ongoing investigations."
Assistant Chief Constable Malcolm Graham said: "Police Scotland is committed to ensuring that all reports of sexual crime committed against children or young people are thoroughly investigated.
"I recognise that it is often a number of years before victims feel able to report such crimes, and would wish to give reassurance that these reports will be treated seriously regardless of the passage of time or status of the perpetrator."
A spokeswoman for the Scottish Government said: "This information referred to in the report was passed to the police and as such this is a matter for Police Scotland."
Among the findings contained in the report was that Savile sexually abused at least five individuals at high-security hospital Broadmoor, in Berkshire, including two patients who were subjected to repeated assaults.
He had keys which allowed him unrestricted access to ward areas within the security perimeter.
One former nurse at the hospital also reported that Savile had claimed to perform sex acts on dead bodies and posed for photographs with the deceased placed in lewd positions.
In the wake of the report, Health Secretary Jeremy Hunt apologised on behalf of the Government and the NHS to the victims of Jimmy Savile's "sickening" sexual abuse in hospitals. He acknowledged they had been "let down badly" and said he hoped honesty and transparency about what happened could help alleviate the victims' suffering.
In January last year a separate joint review by the Metropolitan Police and children's charity NSPCC into allegations of abuse by Savile, found 214 criminal offences, including 34 rapes, had been carried out by the former Top of the Pops presenter across the UK between 1955 and 2009.
The report said Savile sexually abused a teenager at a hospice and he also committed 14 offences at schools across the country, partly after children had written to him for his popular BBC series Jim'll Fix It.