At the start of 2016, Liverpool paid £5.1m to sign Grujic from Red Star Belgrade but the Serbian has only made three starts and 11 substitute appearances for the Merseyside giants in all competitions.
Grujic ended this season on loan at Cardiff City, with the 22-year-old making 13 appearances and helping the Bluebirds earn promotion to the Premier League.
The Welsh outfit are allegedly keen on securing another deal for Grujic but according to Goal.com, Fulham are also ready to offer the player the opportunity of first-team football next season.
Like Cardiff, the Cottagers have just earned promotion out of the Championship and head coach Slavisa Jokanovic appears hopeful of working with his compatriot.
With Naby Keita and Fabinho having already joined Liverpool ahead of next season, Grujic is unlikely to make the breakthrough at Anfield and Jurgen Klopp could potentially welcome the possibility of him earning regular action in the top flight.