The Egypt international has scored 125 in 203 matches - his Premier League record is an impressive 95 in 145 - and while he is some way short of Rush's total of 346 the Welshman thinks his achievements put him in the category of club great.
Salah has two years left on his existing contract and the club are keen to tie him down to longer deal, having already secured Trent Alexander-Arnold, Fabinho, Alisson Becker and Virgil Van Dijk for the long-term.
Rush believes the forward can fulfil his ambitions at the club having won the Champions League, Premier League and FIFA Club World Cup in the last couple of years.
"I know his mentality and he will be going out there to prove to everyone 'I am the best player in the world'," Rush, speaking at a Soccer Aid event, told the PA news agency.
"You've got Ronaldo, Messi and Neymar and you want to be up there with the best.
"(Luis) Suarez and (Fernando) Torres were quality players but they were only at the club a few years.
"Salah will go down as one of the main legends if he carries on doing what he's been doing for the next two years, hopefully even longer.
"I'd hope for Mo Salah to stay there for a good few years. He was our top goalscorer last season and was one of the main players in assists.
"People say he is greedy but he's not greedy, when you play in that position you have to be a bit like that.
"I think he will be looking to improve. It's difficult but you have to give yourself targets, you have to look at trying to improve yourself every year.
"If I scored 30 one season I'd want to score 35 the next."
For a couple of years Salah, Roberto Firmino and Sadio Mane formed arguably the best front three in world football but Lionel Messi's move to Paris St Germain to join Neymar and Kylian Mbappe means the Ligue 1 side have taken that honour, on paper at least.
Rush, however, still believes the options available to Reds manager Jurgen Klopp are enviable.
"I think Liverpool's front four, now Diogo Jota has come in, are a bit more experienced," he added.
"Yes, Paris St Germain have got three top-class players but will they gel? That is the big difference.
"The good thing about Firmino, Salah and Mane is they have a football brain on them. Sometimes you'll see Salah on the right, sometimes on the left and sometimes in the middle and when he makes that run the other two adjust.
"It will be interesting to see if PSG's front three can do that. I think Liverpool's front three plus Jota has delivered.
"They are quality players. If you marked me and Kenny (Dalglish) out of the game you had a chance of winning the game.
"But you have to keep them three quiet but there is always one that shines so to play against Liverpool is very difficult."
:: Soccer Aid for UNICEF 2021 takes place on Saturday 4th September at the Etihad Stadium. The money raised from this year's game will help UNICEF fight back against the Covid-19 pandemic by helping to deliver 2 billion vaccines worldwide. Tickets are available at: www.socceraid.org.uk/tickets