The force has now established a major inquiry team to deal with the referrals.
They have come via the children's charity NSPCC, the Scottish Football Association and directly to police.
The police said they were unable to say how many individuals were affected until they had examined each referral.
The force said every complainer was being spoken to by trained officers from the Child Protection unit.
This unit, along with other experienced detectives, will be carrying out a major inquiry and police have appealed to anyone who has been abused, or had knowledge of abuse taking place, to come forward.
Assistant Chief Constable John Hawkins has told BBC Scotland about the reports reaching them.
He said: "They've come from people with information, but also survivors who've come forward to report child sexual abuse in football.
"When we receive these reports a trained officer will get in contact with the reporter and will arrange to meet them at a convenient time - as soon as possible - in order that we can understand what exactly has happened."
Police forces across the UK, including Police Scotland, are investigating more than 80 potential suspects and 98 clubs in the abuse scandal that has rocked the game.
The Scottish Football Association announced on Tuesday that it was setting up an "independent review" of child abuse allegations in football.
Mr Hawkins said Police Scotland had reacted appropriately to the volume of reports being made to them.
He added: "We've launched a major investigation because of the significant number of referrals that we have received.
"That operation is staffed by officers from our national child abuse investigation unit.
"These are specialist detectives experienced in investigating recent and non-recent child abuse inquiries and they will work with detectives from our public protection units around the country.
"These officers will investigate every report we receive of child sexual abuse in Scotland."
On Friday, a BBC Scotland investigation revealed that former youth coach and referee Hugh Stevenson was allowed to carry on working in football for several years after being reported to police and the SFA over child sex offences.
And Jim McCafferty, a former youth coach who was the kit man for Celtic, Hibernian and Falkirk was arrested in Belfast after allegations were made against him.
Separately, allegations have also been made against coaches who were formerly involved with clubs including Motherwell, Partick Thistle and Rangers.
The allegations involve incidents said to have happened between the 1970s and early 1990s.
The BBC has also learned that about 2,500 coaches are working in youth football in Scotland without having full background checks.
Of the 15,385 coaches registered with the Scottish Youth Football Association (SYFA), 2,500 have not had Protecting Vulnerable Groups (PVG) clearance.
Police Scotland has received 109 referrals involving cases of child sexual abuse in football since a helpline was set up three weeks ago.