The broadcaster claimed he had suffered abuse during his time at Edinburgh Academy in the 1970s.
The force said the investigation was “live and ongoing” and a dedicated team was looking into the claims.
A spokesperson said: “While the investigation of child abuse, particularly non-recent offences, can be complex and challenging, anyone who reports this type of crime can be assured that we will listen and we will investigate all reports, no matter when those offences occurred or who committed them.
“If you have suffered abuse, or you know anyone who may have been the victim of child abuse, then please call Police Scotland on 101.”
I was badly beaten up at school by a teacher who was a leading light in the scripture union
Nicky Campbell made the claim on BBC Radio 5 Live
Campbell made the claims during an episode of his BBC podcast Different.
He said the experience had a “profound effect on my life”.
Later, speaking on his BBC Radio 5 Live show, the broadcaster said: “I was badly beaten up at school by a teacher who was a leading light in the scripture union.
“My mother took it as far as she could and got a grovelling apology from [the man involved], but was essentially stonewalled and it was hushed up by the school.
“Those were different times and that has stayed with me all my life.”
The school said it “deeply” regrets the allegations and “wholeheartedly” apologised to those involved.
In a previously released statement, the school said: “We have worked closely with the relevant authorities including Police Scotland with their inquiries and would like to provide reassurance that things have dramatically changed since the 1970s.
“The Academy has robust measures in place to safeguard children at the school with child protection training now core to the ethos of the Academy.”