The governing body said the investigation is continuing into the gesture, which is regarded by some in France as having anti-Semitic connotations.
A statement from the FA said: "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. The FA will be making no further comment at this time."
The FA is not naming the expert, who was appointed by its governance division to provide detailed knowledge and opinion about the gesture, which was made by West Brom striker Anelka after scoring against West Ham on December 28.
The 'quenelle' - which translates literally as 'dumpling' in English - 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, who has been prosecuted for anti-Semitism, and the salute has been described by opponents as an inverted Nazi salute.
Dieudonne, who has been prosecuted by the French government for insulting the memory of Holocaust victims and holding anti-Semitic views, was banned from performing in Nantes on Wednesday.
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.
He said he made the gesture to show solidarity with his friend Dieudonne.