The club president is "convinced" the promising France centre-back has the potential to be the most senior figure in the dressing room at the Santiago Bernabeu
The 22-year-old moved to the Santiago Bernabeu from Lens in August 2011 and has made 15 appearances in La Liga this season.
Varane has Sergio Ramos and Pepe for competition at centre-back but has not struggled to hold down a first-team place despite his relative inexperience and Perez forsees a significant long-term role for him at the club.
"I am convinced that Varane will one day be captain of Real Madrid," Perez told France Football.
"He is adored by the fans. Not only is he is a great player, he is also a great person."
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While Varane is poised to be a key member of France's squad at Euro 2016 this summer, his club team-mate Karim Benzema's participation is in doubt.
Benzema has been suspended from representing his country pending the outcome of a criminal investigation into the alleged blackmail of Lyon midfielder Mathieu Valbuena, but Perez gave the striker his full backing.
"We have supported him [Benzema] and we will always support him," Perez added. "Having spoken to him, I'm sure that it is impossible to think that he did something bad.
"He is a good lad and I am absolutely convinced that, above all, his intention was to help his friend Mathieu Valbuena."