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."
"An investigation has been launched. There have been no arrests and enquiries continue," they said in a statement.
The English players' union boss Gordan Taylor confirmed this week that Clattenburg is facing allegations he used racist language.
