The Cottagers had previously attempted to sign the left-back last season, but were met with resistance from both the club and the player, however they have doubled their efforts ahead of their promotion to the Premier League.
Warnock is hoping to strengthen his side as they also prepare to enter the English top-flight, but may struggle to compete with the wages offered by other clubs in a similar position.
Aside from Bennett, Fulham are trying to secure several of their loan signings, including Chelsea's Tomas Kalas and Luas Piazon, and Newcastle's Aleksandar Mitrovic and Southampton's Matt Target.