The club said it was aware of a video, widely circulated on social media, which appeared to show racial gestures during the second half of the clash at City’s Etihad Stadium.
Officials from City are working with Greater Manchester Police to identify any individuals responsible. They are also investigating objects thrown onto the pitch, notably at United’s Brazilian midfielder Fred.
“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 statement after United won the derby 2-1.
United manager Ole Gunnar Solskjaer said he was aware of the incident. “I’ve seen it on the video, it was Jesse [Lingard] and Fred and the chap must be ashamed of himself,” he told the BBC’s Match of the Day program.
“It’s unacceptable and I hope he will not be watching any football any more.”
Football around the world has experienced an increasing number of racist incidents in recent years.
“The fact it is still happening is not good enough,” he was quoted as saying by the BBC.
“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.”
Anti-racism body Kick It Out said: “We hope swift action is taken to identify the offenders.”
In Italy this season, Inter Milan’s Romelu Lukaku was the subject of monkey chants from Cagliari fans and Brescia striker Mario Balotelli said he had experienced racial abuse by opposition Verona fans.
Both incidents were met with meager penalties — Verona was handed a one-match partial stadium closure and Cagliari escaped any serious punishment.
In the Manchester derby, Anthony Martial added to Rashford’s opener twith a smartly taken shot inside 30 minutes before Nicolas Otamendi pulled back a late consolation with a header from Riyad Mahrez’s corner.
City, which has won the last two Premier Leage titles, is third in the table, 14 points adrift of leader Liverpool. United is eight points further back in fifth.