A total of 28 children under the age of 15 were referred to the UK Government's Prevent scheme over the course of 2017-2018, with a further 34 people aged between 15 and 20 also referred.
The numbers are up, with eight people under-15 and 23 aged between 15 and 20 referred to the scheme in the previous year.
Prevent was set up to "stop people becoming terrorists or supporting terrorism", Police Scotland said.
North of the border, the strategy aims to tackle all forms of violent extremism and terrorism, including international, Northern-Ireland related and extreme right-wing groups.
The Police Scotland figures show a total of 104 people were referred to Prevent last year, up from 59 in 2016-17.
In 35 cases, it was found no further action was needed, while 66 referrals were passed on to other agencies. Three cases are said to be "ongoing".
According to the police document, 40 of the referrals came from education bodies, including schools, universities and colleges.
The bulk of the concerns focused on "international extremism", with "extreme right-wing" also given as a main reason for referring someone.
The report acknowledged a rise in "extreme right-wing groups" across the UK and stated an increased awareness among the public reflected the heightened number of concerns.
Men made up the majority of the referrals at 96, with eight women being referred.
Assistant Chief Constable Steve Johnson said: "The Prevent strategy in Scotland is delivered through a multi-agency approach which seeks to safeguard individuals vulnerable to radicalisation.
"All partners and communities can have trust and confidence in the process and the latest publication of Prevent referral data is an important step in increasing the transparency of this vital activity."