The Ivorian international has developed a reputation as one of the best midfielders in Ligue 1 since joining Nice from Portuguese side Paços De Ferreira in 2015.
Seri has been linked with moves to the likes of Barcelona, Arsenal and PSG. With the former and latter sending scouts to watch the diminutive midfielder in action against Troyes last Friday.
However, owing to Barca's purchase of Paulinho and given PSG's extravagance in the market already this summer, a deal may be beyond them, whilst Arsenal are unable to make a move until they sell some of their fringe players, leaving Liverpool in the driving seat.
The midfielder excelled last season, making 34 appearances in Ligue 1 and recording seven goals and nine assists as Nice finished in third place.
The 25-year-old is believed to be available for €40m, which is well within Liverpool's budget, given that they saw an offer of €70m for RB Leipzig's Naby Keita turned down earlier this summer.
Although, the sale of defender Dalbert to Inter for €20m means that Nice are in no need to sell the player and will thus not let him go for a penny below their asking price.
The French side are also likely to wait until they have played their Champions League qualifier against Napoli before making any hard and fast decisions over the player's future.