The Santiago Bernabeu chief insists that they must continue to work hard in order to realise Carlo Ancelotti's "spectacular" dream for the club
The 23-year-old was presented to the media at Santiago Bernabeu on Saturday following his €32 million [£27.6m] move from Real Sociedad, but Perez suggested his would not be the last arrival of the summer as they strive to build a team to match Carlo Ancelotti's vision of "spectacular" football.
"We continue to work and look for the spectacular football that Carlo Ancelotti wants," Perez told reporters. "We have an exceptional squad and a team of immense quality, but we are convinced we must keep working to continue being the best club in history.
"Today, a fantastic player arrives at Real Madrid, one who has become one of the great pearls of our game. Asier has been hailed as one of the best players of the [Spain] Under-21 side and he shares the dressing room with many players who are part of that team."
Illarramendi will hold a press conference later on Saturday after being presented to supporters on the Santiago Bernabeu pitch.