The FA was criticised by campaign group Kick It Out earlier this week for their initial silence on the matter and for not acting quickly enough after, and called for "an urgent decision" to be made.
The FA has so far made no comment other than to state an independent expert was aiding its probe, and there would be no further update until January 20 at the earliest.
From the Baggies' perspective, the club are anxious to know whether Anelka will be charged with making an anti-Semitic gesture, as his goal celebration in the 3-3 draw at West Ham on December 28 has been deemed by some.
There is a possibility of the Frenchman being handed a lengthy ban should he be charged and found guilty, denying them a key striker in their bid to avoid relegation this season.
With the club selling Shane Long to Hull for a fee that could rise to £7million, the potential loss of Anelka would further weaken new head coach Pepe Mel's forward options, and with the clock ticking towards the closure of the transfer window at the end of the month.
Albion's sporting and technical director Richard Garlick said: "There are investigations ongoing, and while they are we won't make any comment."
Pressed on the matter, however, Garlick added: "People can speculate there is a turn of events where we might be two strikers down.
"It would help everyone for that investigation to be over as soon as possible, and the FA have given their statement about how quickly they will do that."