Anelka provoked controversy late last month when he celebrated the first of his two goals in a 3-3 draw at Upton Park with the gesture, which has anti-Semitic connotations in his native France.
The former Arsenal, Real Madrid and Chelsea man initially defended his actions, insisting that the celebration was meant as a tribute to anti-Zionist comedian Dieudonne M'bala M'bala, although he has since apologised 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 late January at the earliest.
A statement read: "The FA can confirm that its investigation into Nicolas Anelka remains ongoing.
"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 20 January at the earliest."
