The big-haired Brazilian looks set to leave the Blues in the coming weeks amid talk that a £40m move to Paris Saint-Germain is close to completion – which will make him the most expensive defender on the planet.
The fee could eventually rise to as much as £50m, and Mourinho is keen to use some of this cash to draft in Champions League winner Varane.
The 21-year-old Frenchman starred for Real as they overcame local rivals Atletico Madrid to lift their 10th European Cup over the weekend, and is widely considered to be one of the best young centre-backs around.
Mourinho worked with the classy enforcer while at the Bernabeu, making him one of his first-team regulars despite his youthful age at the time.
Los Blancos are believed to be far from willing to sell their promising starlet, yet Chelsea could make them a financially rewarding offer to offload him.
The Blues are also set to get Kurt Zouma in this summer, with the powerful Saint Etienne man having been away from the club on loan since his switch to west London was agreed in January.