The FA launched a probe following Anelka's gesture at Upton Park on 28 December - an action which, it was alleged, had anti-Semitic overtones.
However, Kick It Out - which is itself supported by the FA, as well as the Professional Footballers' Association (PFA) and the Premier League - believes that the governing body could inject some urgency into their enquiry.
A statement read: "Following a gesture made by West Bromwich Albion striker Nicolas Anelka on 28 December, the FA announced that it would instigate an investigation.
"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.
"While an investigation is being undertaken, it is protocol for Kick It Out not to comment on matters which might prejudice the outcome and be unfair to any party or individual involved in this process.
"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.
"While Kick It Out can and is prepared to be condemnatory, where it is appropriate, it strives to ensure that it works to counter the prejudices which lead to discrimination and hatred. The campaign will certainly make known its comments on this case at the conclusion of the FA's investigation.
"The campaign is totally opposed to acts of antisemitism and has no hesitation in condemning such actions. Kick It Out continues to press for an urgent decision."
Anelka 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.