Both players were fined €500k each after they were involved in a serious incident during Thursday’s training session that reportedly led to captain Valverde being admitted to the hospital.
In a statement released by the club, Madrid said the duo appeared before the investigator assigned to the case and expressed “complete remorse” for their actions.
"Real Madrid C. F. announces that, following the events that led to the disciplinary proceedings initiated yesterday against our players Federico Valverde and Aurelien Tchouameni, both appeared today before the investigator assigned to the case,” a statement from the club read.
“During their appearance, the players expressed their complete remorse for what happened and apologised to each other.
"Furthermore, they extended their apologies to the club, their teammates, the coaching staff, and the fans, and both have made themselves available to Real Madrid to accept whatever sanction the club deems appropriate.
"Under these circumstances, Real Madrid has decided to impose a financial penalty of five hundred thousand euros on each player, thereby concluding the corresponding internal procedures. "
Real said on Thursday that Valverde had been diagnosed with head trauma and would rest for 10 to 14 days.
The club confirmed Valverde was "at home and in good condition" after the incident, which unfolded a day after an earlier altercation between the pair.
Valverde posted a statement on social media on Thursday, apologising to the club and supporters, but denied that things had got out of hand with a teammate and said that, during "an argument" , he had "accidentally knocked over a table".
The gravity of the episode prompted an emergency meeting attended by senior club officials, with no players leaving the training ground for over an hour as Real sought to contain an escalation of tensions that has left nerves frayed and the squad divided.
The unrest followed another reported flashpoint involving defender Alvaro Carreras earlier this week, adding to the strain on the squad.