Liverpool travel to Goodison Park for the first time under Brendan Rodgers and on the back of consecutive wins.
Rush, who is Liverpool's all-time leading goal scorer, believes that it will be a tough game for Rodgers's men.
"Everton have started the season very well and are favourites to win, even if Steven Pienaar is suspended and [Marouane] Fellaini may miss out," Rush said in the Daily Express.
"This is a game where mental toughness comes in. It's a new experience for Brendan, but people like Steven Gerrard and Jamie Carragher will be going around telling the other players 'we have to win it for our supporters.'"
Liverpool take on Everton this afternoon at 1.30pm.