The Egyptian is being linked with a move away from the club in the summer, and if they fail to finish in the top four that speculation may intensify.
However, Rush believes Salah is at home at the Anfield club, and can fulfill his ambitions with Jurgen Klopp's team.
"I think Liverpool has a broad appeal," Reds icon Rush has told HorseRacing.net.
"Teams like Barcelona and Real Madrid have the history, same as Liverpool and Manchester United, and for him to be linked to any of those clubs means that he is one hell of a player but for me I think Mo Salah will stay at Liverpool.
"The fans love him, he loves the fans. The Premier League is the hardest and the most demanding league in the world.
"When you look at Barcelona and Real Madrid, Juventus and Milan in Serie A, where else do you get a league where the bottom team beats the top, like Sheffield United beating Manchester United? That could only happen in the English Premier League."