The Sun says Real are aiming to snap up Vickers in January for £1m plus bonuses.
Arsenal have also been watching the teen this season.
The 18-year-old forward only turned pro with The Royals in July. He made his debut in August and has since made 17 appearances and scored three goals.
Real plan to sign Vickers to work with Raul's Castilla team with the aim of developing him for the seniors.