Anelka scored twice as West Brom came from behind to earn a point at Upton Park, but a gesture made by the 34-year-old after his first goal could land him in trouble.
The action - known in France as the 'quenelle' - is said to have anti-Semitic connotations and a spokesperson from FA has told Perform that an investigation into the incident is forthcoming.
Speaking after the game, West Brom's caretaker manager Keith Downing said he did not expect the player to receive any punishment, and Anelka looked to pre-empt any charge by explaining the issue on his official Twitter page.
He wrote: "This gesture was just a special dedication to my comedian friend Dieudonne (M'Bala M'Bala)."
However, he may face punishment and the FA will have the full support of the Kick It Out campaign, as they released a statement following the incident.
It read: "Kick It Out has been made aware of West Bromwich Albion striker Nicolas Anelka making an alleged gesture, known as the 'quenelle', which carries anti-Semitic connotations during today's (28 December) fixture against West Ham United at Upton Park.
"The campaign is in contact with partners in England and France regarding this matter and has offered its support to The Football Association in any forthcoming investigation."
