City coach Mikel Arteta was widely reportedly to have suffered a cut to the head in the melee that involved up to 20 players and members of staff of the two teams.
Britain’s Press Association reported United boss Jose Mourinho was embroiled in a “heated exchange” with City players after his side were beaten 2-1 by the Premier League leaders.
United were unhappy about City’s whooping and cheering following a win that extended their league lead over second-place United to 11 points.
Neither club has commented on the incident and have not responded to CNN’s request for comment.
Greater Manchester Police told CNN it hadn’t been contacted by either club or any individual over the bust-up.
Mourinho had water thrown at him and the Press Association reported that United’s manager was involved in a “confrontation” with City goalkeeper Ederson.
Usually at Old Trafford one police officer is always positioned outside the referee’s dressing room, which is very close to the visiting team’s locker room.
Both clubs have until December 13 to respond to the Football Association.
‘Better in all departments’
David Silva gave City the lead before Marcus Rashford equalized just before halftime, but Nicolas Otamendi grabbed a second-half winner after capitalizing on Romelu Lukaku’s error.
Mourinho was particularly unhappy that referee Michael Oliver booked United midfielder Ander Herrera for diving after a challenge by Otamendi, rather than giving the home team a penalty.
“It is a huge penalty in a crucial moment of the game,” Mourinho told reporters.
City boss Pep Guardiola said: “We won because we were better, in all departments.”
It’s not the first time that there has been a post-match melee at Old Trafford.
In October 2004, then United manager Alex Ferguson was hit by pizza thrown by then Arsenal midfielder Cesc Fabregas in an incident dubbed “The Battle of the Buffet.”