Images appear to show a fan making gestures towards Fred during United's 2-1 win at the Etihad Stadium on Saturday.
Sources have told ESPN FC that both Fred and Lingard reported they felt they had been targeted with abuse of a racial nature to United officials after the final whistle.
A statement issued by Manchester City on Saturday evening 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."
City also working with Greater Manchester police to identify fans who threw objects on to the field when Fred was taking a corner in front of them.
Fred appeared to have been struck by at least one object while taking a corner.
"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," added the City statement.
United manager Ole Gunnar Solskjaer admitted after the game the incident in the 67th minute had "affected" both his players.
"I've seen it on the video, it was Jesse and Fred and the fella must be ashamed of himself," said Solskjaer. "It's unacceptable and I hope he will not be watching any football anymore.
"Fred and Jesse were taking a corner and I've seen the video, heard from the boys, we keep talking about it every bloody week, but he's been caught on camera, he should never be let in a football ground again, and it doesn't stop unless it'll have consequences. "It affected both Fred and Jesse, you'll see the video if you want to, but sometimes you feel sorry for these people, I'm sure they will feel sorry for them."
Anti-racist campaign group Kick it Out said on Twitter that they had received a number of complaints about incidents at the game.
"We have been inundated with reports of alleged racist abuse from a number of individuals during this evening's Manchester Derby," said the group. "We will be contacting both clubs to offer our support and hope swift action is taken to identify the offenders."