The picture, published in more than one national newspaper, seems to show a fan making a 'monkey' gesture towards a United player during Wednesday night's Capital One Cup clash between the two clubs at Stamford Bridge.
Police are also aware of the photograph but have not yet received a formal allegation on which to open a criminal investigation.
A statement from the 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.
"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."
This latest issue comes at a sensitive time for the club having lodged a formal complaint with the Football Association against referee Mark Clattenburg for using "inappropriate language" towards midfielder Jon Obi Mikel. That matter is also being investigated by police.
Concerning the photograph, a spokesperson for the Met told Press Association Sport: "We are aware of the picture but we have not received a complaint."
Earlier this year Chelsea banned for life a supporter who admitted racially abusing former player Didier Drogba.
Another high-profile incident involving the club concerned captain John Terry, who is currently serving a four-match ban after being found guilty of racially abusing QPR defender Anton Ferdinand by the FA.