The club described it as the biggest internal fine in its history.
The incident reportedly left Valverde hospitalised after suffering what was described as a traumatic brain injury.
While Tchouameni returned to training on Friday, Valverde is expected to miss this weekend’s LaLiga clash against Barcelona.
In a statement, Real Madrid said both players appeared before the club’s investigating officer and expressed regret over the incident.
“Real Madrid C.F. announces that, following the events that led to the opening of disciplinary proceedings yesterday against our players Federico Valverde and Aurélien Tchouaméni, both appeared today before the investigating officer.
“During their appearance, the players expressed their sincere regret for what happened and apologised to each other.
“They also conveyed their apologies to the club, their teammates, the coaching staff and the fans, and both offered their full cooperation to Real Madrid, ready to accept whatever sanction the club deems appropriate.
“In light of these circumstances, Real Madrid has decided to impose a fine of five hundred thousand euros on each player, thus concluding the corresponding internal procedures,” the statement read.
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