The French striker has denied the salute had any anti-Semitic connotations but he caused a storm in his native France by the gesture during the 3-3 draw with West Ham.
West Brom say they accept the celebration has caused some offence and that Anelka has been asked to refrain from doing it again - and that he has agreed to do so.
A club statement said: "The club fully acknowledges that Nicolas' goal celebration has caused offence in some quarters and has asked Nicolas not to perform the gesture again. Nicolas immediately agreed to adhere to this request."
The 'quenelle' salute was made famous in by French comedian Dieudonne M'Bala M'Bala, who has been prosecuted for anti-Semitism. The European Jewish Congress has demanded action against Anelka, claiming it is an inverted Nazi salute.
French minister for sport Valerie Fourneyron also condemned the gesture as "shocking" and "disgusting".
Anelka has stated on Twitter it was nothing more than a "special dedication" to his friend Dieudonne.
The Football Association is continuing to investigate and West Brom are pursuing their own inquiries.
The West Brom statement added: "Nicolas was asked to explain his goal celebration by caretaker head coach, Keith Downing, within minutes of the game finishing at West Ham. Nicolas said that he performed the gesture to dedicate his goal to a friend and vehemently denied having any intention to cause offence.
"Upon reporting for training this morning, Nicolas was asked by sporting and technical director Richard Garlick to give a full explanation about his goal celebration, during which he again strongly denied intending to cause offence.
"The club is aware that the Football Association is investigating the matter and has offered its full co-operation. The club will continue to make its own inquiries - a process which will remain confidential between the club and Nicolas.
"Nicolas is eligible for matches whilst the FA carries out its investigation. Therefore, Nicolas will remain under consideration for first-team selection whilst The FA and club continue their enquiries."
Pictures have also emerged of two other French players, Samir Nasri and Mamadou Sakho, performing the gesture.
The photograph of Manchester City playmaker Nasri was taken outside the club's Carrington training base and is thought to date back to November.
It is understood that Nasri was unaware of the anti-Semitic or political connotations of the gesture and did it because of its popularity in France.
City manager Manuel Pellegrini refused to comment on the matter when asked at a press conference to preview his side's New Year's Day game at Swansea.
Pellegrini said: "I don't know what you are talking about. I haven't seen it.
"I read that something happened with Anelka but I don't know anything about Samir. I can't talk about something I haven't seen."
Liverpool defender Sakho said in November that he was tricked into performing the gesture.
He wrote on his Twitter account: "This photo was taken six months ago, I did not know the meaning of this gesture, I got trapped!"
On Monday, a Liverpool spokesperson said: "Mamadou Sakho has explained that when posing for the photo, taken over six months ago, he had no knowledge of any meaning or significance attached to the gesture."