" Real Madrid C. F. would like to inform you that, following the events that took place during first-team training this morning, it has been decided to open disciplinary proceedings against our players Valverde and Tchouameni," announced the Merengue club.
Once the internal procedures have been completed, the club will announce any sanctions at a later date.
Real Madrid midfielder and captain Valverde was admitted to hospital after a clash with his French teammate Tchouameni, several Spanish media reported on Thursday.
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The Uruguay international, who was accompanied by Madrid coach Alvaro Arbeola, was admitted to hospital near Real's training centre for several stitches after sustaining a facial injury, according to Spanish media reports.
In a further statement, Madrid said: "After the tests carried out today on our player 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, as indicated by medical protocols for this diagnosis."
The two players, who first clashed during training on Wednesday, clashed again on Thursday during and after the training session, according to the same sources.