Real Madrid announced on Friday they had fined Aurelien Tchouameni and Federico Valverde €500 000 each after a training ground clash that left the latter needing hospital treatment.
The club did not impose any sporting sanctions on the two players, saying in a statement that the fine "thereby concludes the internal procedures" launched against them.
Valverde will miss Sunday's Clasico against Barcelona as a result of the head injury he suffered during the altercation. The club said he would be out for up to two weeks.
Tchouameni took part in training on Friday and could feature at Camp Nou this weekend.
Madrid said both players "expressed their complete remorse for what happened and apologised to each other" while taking part in a club investigation.
"They extended their apologies to the club, their teammates, the coaching staff, and the fans, and both have made themselves available to Real Madrid to accept whatever sanction the club deems appropriate," read a club statement.
Uruguay international Valverde was accompanied to the hospital facility near the club's Valdebebas training complex by Madrid coach Alvaro Arbeloa, according to Spanish reports, which said the player needed stitches to treat a facial wound.
Uruguay midfielder Valverde sought to downplay the severity of the altercation with the France midfielder.
"The strain of the competition and frustration caused the situation to escalate," Valverde wrote on social media, expressing regret at the media coverage of the incident.
"I accidentally hit a table during the argument, causing a small cut on my forehead that required a routine visit to the hospital," he said.
"At no point did my teammate hit me, and I didn't hit him either."
Tchouameni admitted that what had happened was far from acceptable.
"Regardless of who is right and who is wrong we should always look for a peaceful resolution to any dispute," said the Frenchman.
"There's no point going on about who did what or who said what and I accept the fine," he said. "Now we have to move forward and play this Clasico and make sure the club does as well as is still possible."
According to reports the two players rowed on Wednesday in training and their argument continued on Thursday during and after the session.
Spanish media reported Valverde refused to shake Tchouameni's hand and later fouled him in Thursday's training session, with the pair scrapping afterwards in the dressing room when the injury occurred.
Tensions are running high at Real Madrid with the club on the verge of a second consecutive season without a major trophy.
Los Blancos trail Barca by 11 points at the top of LaLiga, with Hansi Flick's side able to clinch back-to-back league titles on Sunday if they do not lose.