In scoring the first of his two goals in Saturday's 3-3 draw at Upton Park, the 34-year-old seemingly performed the "quenelle" salute made famous in his homeland by the comedian, Dieudonne.
The gesture has been linked to anti-Semitism in France but Anelka took to Twitter to defend his actions, saying it was "just a special dedication to my comedian friend Dieudonne".
The salute, though, is the subject of fierce national debate and the French sport minister Valerie Fourneyron condemned the gesture as "shocking" and "disgusting".
Dr Moshe Kantor, president of the European Jewish Congress, wants Anelka to be banned. "It is sickening that such a well-known footballer would make such an abusive and hateful gesture in front of tens of thousands of spectators," he said.
"There should be no room for such intolerance and racism in sports and we expect that the English Premier League officials as well as the police will give Anelka the appropriate punishment. Furthermore, we hope that this gesture is banned in all places of Europe where Nazi salutes are banned. Merely inverting the traditional Nazi salute should not allow anti-Semites to spread and display their hate with impunity."
It is understood the Football Association will be looking into the gesture, while anti-racism campaign group Kick It Out has offered its assistance.