The duo were reported to have fought at the end of Thursday’s training session following a previous incident on Wednesday.
Valverde went to hospital with a head injury after a dressing room row with Tchouameni but denied the pair “came to blows”.
Real said the two players expressed “complete remorse” during an internal disciplinary process which has resulted in hefty financial penalties.
A club statement read: “Real Madrid CF 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.
“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 team-mates, the coaching staff and the fans, and both have made themselves available to Real Madrid to accept whatever sanction the club deems appropriate.
Scrappers: Aurelien Tchouameni and Federico Valverde
“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.”
It is the latest in a string of recent incidents at the club, with defender Antonio Rudiger reportedly apologising to the squad for another training-ground clash.
Star striker Kylian Mbappe was then reported to have been involved in an angry exchange with a member of the club’s coaching staff who was acting as an assistant referee in a training match.
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Mbappe, who is recovering from a hamstring injury and is a doubt for this weekend’s game, has also faced criticism for a trip to Italy with his partner.