The 24-year-old arrived at Madrid from Real Sociedad last summer, but struggled to hold down a first-team spot during his first season at the Bernabeu.
It had been suggested that Illarramendi would seek a loan move this summer, but Perez has quelled such claims.
"I think Illarramendi will be with us for many years and he will do very well," Perez told Onda Cero.
"I don't know why I would have signed him otherwise."
Madrid won a Champions League and Copa del Rey double this season.