French midfielder Sylla - who plays for Ligue Two side Clermont Foot - will cost Villa around £1.7million.
Lambert and chief executive Paul Faulkner were trying to seal the deal on Wednesday evening.
Lambert has identified midfield as an area he needs to bolster and he was beaten to the signature of Moussa Sissoko by Newcastle.
Sylla, 22, is a defensive-minded player who will provide protection for Lambert's young back four.
Lambert spent £23million on new signings last summer but has only limited funds available now.
But he indicated earlier this week that he would like to bring in "one or two signings" and is on the verge of securing his first addition to the squad.