The Metropolitan Police received a complaint after the fan was pictured in British media, and said it is now investigating the "alleged racist behaviour at Stamford Bridge."
Chelsea is also investigating the incident during the 5-4 victory over United.
"The club will be examining all available footage and asks anyone who can identify the individual to contact the club," the Premier League leaders said in a statement. "Chelsea FC is committed to removing all forms of discrimination and if we have sufficient evidence we will take the strongest possible action, including supporting criminal prosecution."
Police are also investigating allegations referee Mark Clattenburg directed a racial insult at midfielder John Mikel during another meeting between Chelsea and Manchester United in the league on Sunday.
Chelsea has not complained to the police but sent a file to the FA on Wednesday as part of a formal complaint.