Cadena Cope reported the development on Thursday. The parents are said to have pestered Pérez repeatedly in recent weeks, with head coach Álvaro Arbeloa also drawn into the dispute. Neither player has been named publicly.
The revelation follows the most turbulent week in the club's recent history. Defender Antonio Rüdiger was reported to have struck left-back Álvaro Carreras during a training session two weeks ago. On Wednesday, midfielder Fede Valverde and French midfielder Aurélien Tchouaméni clashed during a training drill at Valdebebas, with the argument continuing into the dressing room.
A second, more serious confrontation between the same two players followed the next day, resulting in Valverde being taken to hospital with a head injury he later attributed to accidentally striking a table during the argument.
Marca described the second incident as much worse than the first, with teammates having to intervene before Valverde was taken for medical attention. Real Madrid confirmed Thursday evening they have opened disciplinary proceedings against both players.
In their official statement, the club said: "Real Madrid CF announces that, following the incidents that occurred this morning during the first team's training session, it has decided to open disciplinary proceedings against our players Federico Valverde and Aurélien Tchouaméni."
A second statement confirmed Valverde's diagnosis.
"After the tests carried out today on our player Fede Valverde by the Real Madrid Medical Services, he has been diagnosed with a cranioencephalic trauma. Valverde is at home in good condition and will need to rest for 10 to 14 days."
Valverde later denied either player had struck the other, insisting the confrontation was a heated argument rather than a physical fight.
French forward Kylian Mbappé has also drawn internal criticism this season, though details of his specific involvement in the fractious atmosphere have not been confirmed publicly.
Real sit eleven points behind Barcelona in La Liga with four games remaining. The Clásico at the Bernabéu on Sunday will be played without Valverde — and in a dressing room that has never looked more fractured.