An incident involving up to 20 players and staff from both clubs after City's 2-1 victory at Old Trafford left Mikel Arteta, one of Pep Guardiola's assistants, with a facial cut.
Both clubs insist no punches were thrown during the clashes and police are yet to open an investigation into the incident, having not received a related complaint.
An officer present at the time of the clashes, which occurred in the tunnel between Old Trafford's home and away dressing rooms, also concluded no criminal offence had taken place.
Manchester United vs Manchester City player ratings
Show all 22
The melee is understood to have started after United manager Jose Mourinho took exception to City's post-match celebrations.
Mourinho and City goalkeeper Ederson are reported to have become involved in a confrontation which then escalated with players and staff of both clubs involved.
The Football Association have asked United and City to submit their observations of the incident before 6.00pm on Wednesday, after which the governing body may decide to take action against both clubs.
On Tuesday, Mourinho and Guardiola addressed Sunday's post-match events for the first time ahead of the midweek round of Premier League fixtures.
The City manager claimed his players did not celebrate excessively after opening up an 11-point lead over their neighbours and nearest challengers and emphasised their respect for their rivals.
“I encourage the players to celebrate the victories, but of course inside the locker room," Guardiola said. “We have huge respect for our opponents - not just for Manchester United, all the opponents.”
Meanwhile, though reluctant to discuss the incident, Mourinho questioned Guardiola's claim that City's celebrations.
“I'm not here to comment on his words,” the United manager said. “The only thing I can say is that for me it was just a question of diversity. Diversity in behaviours, diversity in education. Just that and nothing more than that.”
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies