Pictures published on the internet appear to show a man making a monkey gesture in the stands during Chelsea's 5-4 extra-time victory at Stamford Bridge.
"Chelsea Football Club can confirm we are investigating an incident involving a member of the crowd at last night's game against Manchester United," the club said in a statement.
"The club will be examining all available footage and asks anyone who can identify the individual to contact the club. 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."
The Metropolitan Police are also investigating after receiving a complaint.
"An investigation has been launched. There have been no arrests and enquiries continue," they said in a statement.
The incident comes three days after the Premier League encounter between the two clubs when Chelsea accused referee Mark Clattenburg of using inappropriate language towards midfielder John Obi Mikel.
The English players' union boss Gordan Taylor confirmed this week that Clattenburg is facing allegations he used racist language.
Chelsea captain John Terry is currently serving a four-match ban for racially abusing Queens Park Rangers defender Anton Ferdinand.
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