The Reds are already thought to be lining up targets for the end of the season, and are ready to snap up stars to take them to the next level if the qualify for the Champions League.
Benzema is seen as just the sort of player to push the club on by Rodgers, who is confident that he can lure the Frenchman to Anfield.
The Real ace is thought to be unsettled at the Bernabeu, and may be allowed to leave if they snap up a new striker thanks to the emergence of starlet Jese Rodriguez.
If Benzema arrives he would form a fearsome front three with Luis Suarez and Daniel Sturridge, which would rank among the very best attacking set-ups on the continent.
Arsenal and Manchester City have also been linked with the ex-Lyon man, but Rodgers is hopeful that he could persuade him to become a part of his Merseyside revolution.
Benzema is likely to cost over £25m, but the potential added income from top level European football could fund such a move.