The Brazilian attacker had been linked with a move away from the Santiago Bernabeu after finding his first-team opportunities limited last season.
However, Perez has insisted that the 31-year-old will get a chance to play under his former AC Milan boss Carlo Ancelotti.
Perez told reporters: "Kaka is a phenomenon, but he got injured in the last World Cup and had to have an operation on his knee.
"Then [Mesut] Ozil arrived and took his place in the team from him, but he is still one of the greats of the game. He is the second most followed footballer on Twitter, after Cristiano Ronaldo.
"He is charming and excites the fans. I'm optimistic that, under Ancelotti, who knows him very well, he can be the same magical Kaka again that we signed."
Kaka signed for Madrid in 2009 for £53.5m.