The 22-year-old joined the Scottish champions from Norwegian outfit Start in 2016 and has featured on 129 occasions for the Hoops in all competitions.
Ajer has contributed four goals in 49 appearances for Neil Lennon's side during the 2019-20 campaign, and the likes of Wolverhampton Wanderers, Arsenal and Leicester City have all recently been linked with his services.
However, according to Mundo Deportivo, Los Blancos are also keeping a close eye on his development as they search for a long-term Sergio Ramos replacement.
Ajer's existing deal at Celtic is due to expire in the summer of 2022, and the Scottish giants are thought to value the defender in the region of £25m.
The Norway international has won three Scottish Premiership titles at Celtic Park.