The former Wales striker says that the Belgian must start to be more selfish if he is to succeed.
The Belgian has scored just seven times in 31 matches so far for the Reds during his debut season on Merseyside after joining the club for a fee of around £32 million from Aston Villa in the summer.
The 25-year-old's performances have received strong criticism from some sections of the Liverpool support since his arrival and his poor form has seen him fail to score so far in 2016.
However, Rush has backed the striker to force his way back into contention for a regular starting spot under Klopp but advised him that he may have to change the way he approaches play if he is to impress.
Rush told Sky Sports: "What he has to do is get in the box, you have to be selfish and score goals. I went seven games without scoring for Liverpool once, and I got dropped even though we'd won five and drawn two.
"I think Benteke has to be a little bit selfish in the box because when you are a striker, you have to be hungry at the right time. Strikers will be judged on goals, not how many you create, and once you start scoring, you start making them as well.
"If you get him inside the box more, he'll get chances because he is a good finisher and I think he can play in the Liverpool system. I think Benteke is a quality player, and I've seen him for Belgium and Aston Villa when he scored some great goals. He just needs a goal."
In his time at Anfield, Benteke has thus far failed to scale the same heights that he did whilst being used as the focal point in three years at Aston Villa.
The Belgian, who arrived in English football from Genk in 2012, boasted a ratio of nearly a goal every other game for the Birmingham side, hitting the net 49 times in just 101 matches in all competitions.
He has found life far more difficult at Anfield, where he has had to compete with Daniel Sturridge, Roberto Firmino, and Divock Origi for a place in attack all season.
Liverpool currently sit eighth in the Premier League table but have another break from league action this weekend as they prepare to face Manchester City in the Capital One Cup final at Wembley on Sunday.