The Football Association has announced it is working with an "expert" in relation to the gesture and will not announce a decision until January 20 at the earliest.
Anelka provoked controversy last month when he celebrated the first of his two goals in a 3-3 draw at Upton Park with the gesture, which is claimed to have anti-Semitic connotations.
The former Arsenal, Real Madrid and Chelsea man initially defended his actions, insisting that the celebration was meant as a tribute to comedian Dieudonne M'bala M'bala, although he has since apologized and vowed not to repeat the action.
Jewish groups have called for the forward to be banned for the celebration, but the Football Association will not making a decision until January 20 at the earliest.
"The FA can confirm that its investigation into Nicolas Anelka remains ongoing," a statement read. "We are working with an appointed expert regarding the salient issues related to the 'quenelle' gesture.
"Whilst this work continues as quickly as possible, there will be no further update until Monday January 20 at the earliest."