It was reported on Wednesday that the two got into a heated argument following a tackle Valverde, 27, made on Tchouameni, 26, during training.
That disagreement continued into the following day and the two took it even further, with Valverde been rushed to hospital in a wheelchair with a head injury.
According to Spanish media, Valverde was accompanied by manager Alvaro Arbeloa and needed several stitches in his face.
Real Madrid have since released a statement confirming that the club have launched disciplinary proceedings against the two midfielders.
“Real Madrid CF announces that, following the events that took place this morning in the first team training session, it has decided to open disciplinary proceedings against our players Federico Valverde and Aurélien Tchouameni,” the statement read.
“The club will announce the resolutions of both cases in due course, once the corresponding internal procedures have been completed.”