A picture of the alleged incident was published in several national newspapers, prompting Chelsea to announce a club probe.
A formal complaint has further been made to the Metropolitan Police, who have begun a criminal investigation.
A statement released by the force on Thursday read: "Police have received a complaint regarding alleged racist behaviour at Stamford Bridge (on) Wednesday, October 31.
"An investigation has been launched. There have been no arrests and enquiries continue."
A statement from the west London club read: "Chelsea Football Club can confirm we are investigating an incident involving a member of the crowd at (Wednesday) night's game against Manchester United.
"The club will be examining all available footage and asks anyone who can identify the individual to contact the club.