The Reds have been strongly linked with the 26-year-old Ivory Coast international, considered as a potential replacement for Emre Can who looks set to leave at the end of the season.
Liverpool are said to have been impressed by Seri's performances since he arrived at Nice in 2015, racking up 97 Ligue 1 appearances.
But now, according to L'Equipe, Chelsea are also pushing hard to land the midfielderand have already enquired with Nice about a summer move.
It has also been suggested that Seri and Nice have an unwritten agreement that will allow him to leave should any club make an offer worth £35million.