Rush knows what it means to be a top class goalscorer for the Anfield club, having led Liverpool to domestic and European titles.
However, he hopes that unlike when he left the club in 1987 for Juventus, Salah stays put.
He told Liverpool Echo: "I'd love him to extend his contract. We don't know what will happen but I'm sure every Liverpool supporter would love that as well.
"I left because I wasn't playing regularly. I just wanted to play football. I didn't want to sit on the bench and that's what you've got to look at. I left here to go to Leeds United because I wanted to be playing week in, week out.
"I looked at (Robbie) Fowler and (Stan) Collymore, they were doing really well together, and I didn't want to be sitting on the bench. I was enjoying my football and that's the reason why I went to Leeds."
He added: "But that was my situation, it depends if you are happy sitting on the bench or not. But Mo's not even got to that stage yet so no, there's not really any reason for him to go yet."