The pair fought at the end of Thursday's training session, following a previous altercation on Wednesday, and Valverde was then revealed to have suffered a head injury that will rule him out of most of Real's remaining games this season.
A club statement read: "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 Aurelien Tchouameni."
But Valverde posted a lengthy statement in response on his Instagram story - without mentioning Tchouameni by name - in which he insisted it was a meaningless fight and the incident had been "blown out of proportion".
The Uruguay midfielder wrote: "Yesterday (Wednesday) I had an incident with a teammate during a training session.
"In a normal locker room, these things can happen and are usually resolved internally without it becoming public knowledge.
"Obviously, there's someone behind this, spreading the story quickly, and with a season without titles where Real Madrid is always under scrutiny, everything gets magnified.
"Today we had another disagreement. During the argument, I accidentally hit a table, causing a small cut on my forehead that required a routine visit to the hospital.
"At no point did my teammate hit me and I didn't hit him either, although I understand it's easier for you to believe we came to blows or that it was intentional, but that didn't happen.
"I'm sorry. I'm truly sorry because this situation hurts me, and this moment we are going through hurts me."
Valverde's description of a "small cut" was in contrast to the club's medical update, which read: "After the tests carried out today on our player Fede 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."
That timescale rules Valverde out of Sunday's Clasico - Real's arch-rivals Barcelona will clinch the title should they avoid defeat - and potentially the remainder of Real's season, which concludes on May 24 at home to Athletic Bilbao.
Real gave no timeline on the disciplinary proceedings.
"The club will provide updates on the resolutions of both proceedings once the corresponding internal procedures have been completed," it said.
It is the latest in a string of recent controversies at the club, with defender Antonio Rudiger reportedly apologising to the squad for another training-ground clash.
It was not the only altercation involving Madrid players during training this week. Defender Álvaro Carreras confirmed on Instagram that he was involved in "an incident with a colleague." He called it a "minor" and "isolated" matter that "has already been resolved."
Spanish media said he and defender Antonio Rüdiger got into a scuffle during one of the team's training sessions.
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.
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.
He arrived back shortly before kick-off in last Sunday's win at Espanyol.
with AP
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