The 26-year-old Ivory Coast international has impressed in France since joining Nice in 2015 for a €1million fee.
Seri was wanted by Barcelona last summer but a deal fell through, and the Spanish giants have since dropped their interest.
Arsenal, Manchester United and Tottenham have all been closely watching the player this season, as they draw up their summer shopping plans.
But, according to The Sun, it is Liverpool who are the frontrunners to secure Seri's signature.
Jurgen Klopp is expecting to lose Emre Can this summer, with the midfielder out of contract and in talks over joining Juventus.