Scotland Yard has formally recorded eight criminal allegations against the star, including two of rape and six of indecent assault.
It has assembled a team of officers to look into the claims.
The move came as the family of the late TV presenter said his ornate £4000 grave in a cemetery in Scarborough, North Yorkshire, was to be dismantled "out of respect for public opinion."
Undertakers were today due to remove the triple headstone, which they only put up on September 20, out of respect to the relatives of others buried there.
In London, Commander Peter Spindler, head of Specialist Crime Investigations at the Metropolitan Police, said the allegations spanned four decades and information so far suggested abuse was on a "national scale".
He said there were 120 lines of inquiry, although some could be duplicates.
He said: "We are getting calls from victims, from witnesses and third parties who believe they know something about it.
"We have formally recorded eight criminal allegations against Savile. Two of those are rape, six of indecent assault.
"These are primarily against girls in their mid-teens, so between 13 and 16, and it spans four decades of abuse."
He said the first dated back to about 1959 but most seemed to be in the 1970s and 1980s.
Scotland Yard has been in contact with ITV and the BBC to gather information. Alleged victims that officers have been talking to have been contacted to see if they will co-operate.
Mr Spindler said: "We believe we will come up with between 20 to 25 victims."
Mr Spindler said police are working to identify anyone who could be subject to criminal investigation. The BBC is not being investigated, he said.
Asked if allegations related to any institutions other than those mentioned in previous claims – the BBC, the Jersey children's home, or the school in Staines – he said he had contacted Stoke Mandeville Hospital, and Leeds Royal Infirmary.
He said: "The pattern of his offending behaviour does appear to be on a national scale."
Meanwhile, Freddie Starr said he would welcome the opportunity to speak to police about "devastating" claims he was involved in groping a teenager.
The comedian, 69, spoke out after one of Savile's accusers alleged in a TV interview that Starr had touched her when she was just 14 when they were both in a BBC dressing room.
Asked whether he had ever fondled an underage girl, he said: "Never. I never have done. I never will do. It's not my style. It's not in me to do that.
"If there's one thing I hate, close to my heart, it is paedophiles."
In Birmingham, Prime Minister David Cameron raised the prospect yesterday that Savile could be posthumously stripped of his knighthood.
Mr Cameron said: "We have something called the Forfeiture Committee.
"It's not chaired or sat on by me, but it is responsible for looking at honours and the removal of honours and obviously they have to do their job too."