The Football Association will investigate allegations of racism after Manchester United players said they were targeted at Manchester City.
City said they were "aware of a video circulating on social media" which appears to show a supporter making monkey gestures towards United players.
They have pledged to issue a lifetime ban to "anyone found guilty of racist abuse".
The FA plans to speak to the clubs, referee Anthony Taylor and the police.
The incident in question happened when United midfielder Fred went to take a corner in the second half.
The 26-year-old Brazilian said it was a shame that such incidents still happen in 2019.
"We are still in a backward society," Fred told ESPN Brazil after the 2-1 win for Manchester United.
"Unfortunately, this is happening in some stadiums. It happened here, it happened in Ukraine with some friends.
"It's sad, but we have to keep our heads up and forget about that. We can't give them any attention because that's all they want. I spoke to the referee after the match, they will do something about it and that's all."
Fred also appeared to be hit by an object thrown at Etihad Stadium.
Anti-racism body Kick It Out says it has been "inundated" with reports of alleged racist abuse after the incidents were captured by television cameras.
"We hope swift action is taken to identify the offenders," Kick It Out said.
More than one United player said they had been abused after the game, with the Old Trafford club reporting their comments to referee Anthony Taylor and Manchester City.
City said they are working with Greater Manchester Police to help them identify any individuals who were involved. Greater Manchester Police said that no arrests had been made but that "enquiries into the incident are ongoing".
"The club operates a zero-tolerance policy regarding discrimination of any kind," City added.
The Professional Footballers' Association welcomed City's prompt response, adding: "Racist abuse is a criminal offence and must be dealt with accordingly."
United manager Ole Gunnar Solskjaer said: "I've seen it on the video and the fella must be ashamed of himself. It is unacceptable and I hope he won't be watching any football any more."
City manager Pep Guardiola said he does not want to see any more alleged racist abuse "happen again".
"It is a battle to fight every day. Unfortunately, it has happened in many places," he said.
United forward Marcus Rashford, who was also playing when England's Euro 2020 qualifier in Bulgaria was overshadowed by racism in October, called for more to be done to tackle the problem.
"The fact it is still happening is not good enough," he said.
"We seem to be speaking about it an awful lot over last six to eight months. Even speaking about it now is not nice.
"The necessary departments need to do the right things to stop it in the game. It is a big negative in the sport and the country."
With United leading 2-0, a number of objects were thrown by supporters in the home end when Fred went to take a corner in the 67th minute.
The Brazilian moved away from the corner flag before going back to take the set-piece.
City midfielder Fernandinho, along with other home players, urged the fans in that corner to calm down.
Play resumed a few moments later once referee Taylor picked up a number of objects in that area of the pitch.