Reports first emerged in the Spanish media on Wednesday that midfielders Federico Valverde and Aurelien Tchouameni had been involved in a heated verbal disagreement during training.
It was later confirmed by Valverde himself and the situation is understood to have escalated on Thursday at Real's training base.
Sources have told BBC Sport the argument continued after training and culminated in Valverde being taken to hospital with a head injury following a dressing room altercation.
Valverde later denied reports the pair had physically fought, insisting the injury occurred after he "accidentally hit a table" during the confrontation.
In a lengthy statement released on Thursday evening, the Uruguay midfielder said he suffered "a small cut on my forehead that required a routine visit to the hospital" and rejected suggestions either player had struck the other.
An emergency meeting was later called involving club president Florentino Perez, members of the coaching staff, head coach Alvaro Arbeloa and captain Dani Carvajal.
Real Madrid subsequently released two statements.
The first confirmed disciplinary proceedings had been opened against both players, adding that the club would provide updates "once the corresponding internal procedures have been completed".
The second was a medical update confirming Valverde had suffered a concussion and would need to rest for between 10 and 14 days, ruling him out of Sunday's El Clasico.
"Clearly, someone here is spreading rumours, and with a season without titles, where Real Madrid is always under scrutiny, everything gets blown out of proportion," Valverde said.
On Friday, Real said the pair had "apologised to each other, the club and their team-mates" and been fined 500,000 euros (£432,037.50) each.
These issues and the team's reltively poor results have meant focus has turned to head coach Alvaro Arbeloa and his apparent inability to maintain control of the dressing room during this turbulent period.
The former Real defender succeeded Xabi Alonso, who lasted six months in the role, in January.
His lack of senior managerial experience was a talking point from the moment he was appointed, with questions raised over the ability of someone who had only previously worked within the youth set-up to manage established stars such as Mbappe and Vinicius Junior.
This week's events have validated those concerns for many supporters.
With only four matches remaining this season for Real, Arbeloa's immediate priority will be to restore stability, refocus the squad and prevent further deterioration in performances and discipline.
While silverware is no longer a realistic target, there remains a need to finish the campaign with some sense of control.
Beyond the short term, however, the focus is already shifting towards the broader structure above him.
There is now pressure on club president Florentino Perez, who faces a major decision after a run of managerial changes that have failed to deliver sustained success.
Three managers. Two seasons. No trophies.
The choice of the next permanent head coach will be pivotal, not only for results on the pitch, but for restoring stability and control over a squad that has appears difficult to manage.
That in turn will repair the image of a club that is receiving uncomfortable public criticism and, they will hope, return them to the top of the club game.
