Manchester City have said they are "aware of a video circulating on social media which appears to show a supporter making racial gestures" during their match against Manchester United.
The club also said they are "working with Greater Manchester Police in order to help them identify any individuals concerned".
The alleged incident occurred in the second half of City's 2-1 defeat at the Etihad as a supporter appeared to make a monkey gesture towards a United player.
The PA news agency understands more than one United player felt they had abuse of a racist nature aimed at them. United reported the matter to City and referee Anthony Taylor.
— Manchester City (@ManCity) December 7, 2019
A City statement after the game read: "Manchester City FC are aware of a video circulating on social media which appears to show a supporter making racial gestures during the second half of the match against Manchester United this evening.
"Officials from the club are working with Greater Manchester Police in order to help them identify any individuals concerned and assist with their enquiries.
"The club are also working with GMP regarding an instance of objects being thrown onto the field of play.
"The club operates a zero tolerance policy regarding discrimination of any kind, and anyone found guilty of racial abuse will be banned from the club for life."
English football's equality and inclusion organisation Kick It Out said it had been "inundated with reports of alleged racist abuse from a number of individuals" during the game.
It added: "We will be contacting both clubs to offer our support and hope swift action is taken to identify the offenders."
United manager Ole Gunnar Solskjaer branded the alleged abuse "unacceptable".
He told Sky Sports: "I've seen it on the video. Fred and Jesse (Lingard) in the corner taking coins from the City fans first of all, so they've got to go away.
"And then there's this not-so-intelligent fella chanting racist abuse towards my two players and I hope City and the authorities will deal with it, because he shouldn't be watching football again. Not live, anyway.
"We have to stamp it out. We keep talking about it every single week and it's not acceptable. He should not be allowed back into a ground. That's how simple it is."
United goalscorer Marcus Rashford said he was not aware of the missile-throwing and alleged abuse until after the game.
He said: "But it can happen and in recent times we have been dealing with it a little bit better but the fact it's still happening isn't good enough.
"There's obviously work still to do in that area but for us it's about the football and trying do our best on the pitch."
The Professional Footballers' Association condemned the alleged abuse, saying on Twitter: "We are disappointed by what appears to be racist abuse from fans at today's Manchester derby. We welcome this prompt response from Manchester City. Racist abuse is a criminal offence and must be dealt with accordingly."