The Football Association (FA) had initially asked both clubs to provide their observations by 6pm GMT on Wednesday.
However, English football's governing body has now extended the deadline to 6pm GMT on Friday for both United and City to hand over their versions of events.
Goal reported on Sunday that both teams were involved in a heated altercation in the tunnel following City's 2-1 victory over rivals United.
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 December 15.— FA Spokesperson (@FAspokesperson) December 13, 2017
City's players were celebrating their victory at the home of their rivals when United boss Jose Mourinho, on his way to a post-match press conference, entered the visiting dressing room.
It is believed that the City players were playing music, cheering and singing after goals from David Silva and Nicolas Otamendi put them 11 points clear at the top of the table.
Sources indicate Mourinho took exception, believing the celebrations to be a lack of respect, and told Pep Guardiola's men to calm down with a comment in Portuguese that sparked an ugly reaction.
Goal understands Mourinho was covered in water during the incident while City coach Mikel Arteta suffered a cut eyebrow.