City extended their advantage at the top of the Premier League to 11 points with a 2-1 victory thanks to goals from Nicolas Otamendi and David Silva.
Reports suggested players and staff from both sides clashed in the tunnel after the match as City celebrated in their dressing room.
The Football Association requested information from both camps, who have now been given additional time to respond.
An FA spokesperson said: "Manchester United and Manchester City have both been given extra time to provide their observations following reports of an incident after Sunday's game. They now have until 6pm on Friday 15 December."
Pep Guardiola was adamant City had done nothing wrong, but refused to comment further when asked about the alleged incident in his press conference.
Meanwhile, United manager Jose Mourinho cryptically suggested the apparent altercation was "just a question of diversity; diversity in behaviours, diversity in education."