Anelka scored twice as the Baggies came from behind to earn a point at Upton Park, but celebrated his first goal with a gesture that is seen by some in France to be an inverted Nazi salute.
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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."
But Dieudonne has attracted fierce criticism in recent years for his right-wing political views, which have featured more and more prominently in his performances.
Meanwhile, the Kick It Out campaign have pledged their support to the FA in a statement: "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."
Anelka would face at least a five-game ban if he were found guilty of making an anti-Semitic gesture.