West Brom striker Anelka made the salute after scoring against West Ham on December 28. It is regarded by some in his native France as having anti-Semitic connotations.
The FA has appointed an expert to work on the case and has said no statement will be made before January 20.
Kick It Out, who initially reported an allegation of anti-Semitism to the FA, said in a statement: "Since then, in spite of requests from Kick It Out for information about the progress of the investigation, there has been total silence from the FA until Friday 10 January.
"The FA then stated that an appointed expert was aiding this investigation and there would be no further update until Monday 20 January at the earliest.
"Kick It Out, like others, is very frustrated at the length of time taken to investigate this issue and, as usual, has taken criticism, particularly from community organisations, who feel deeply and rightly aggrieved by the gesture and want to know why the campaign has not made more of a public condemnatory statement of Anelka's actions.
"The campaign will certainly make known its comments on this case at the conclusion of The FA's investigation. Kick It Out continues to press for an urgent decision."
The 'quenelle' is a straight-arm salute with one arm pointing downwards and the other hand across the chest.
It the was brought to prominence by French comedian Dieudonne M'Bala M'Bala, a friend of Anelka's who has been prosecuted for anti-Semitism, and the salute has been described by opponents as an inverted Nazi salute.
Anelka has agreed not to perform the celebration again after West Brom conceded the gesture had caused offence but the French striker has denied the salute had any anti-Semitic connotations.