The Reds' record goalscorer insists the club's high standards of the past must be met - something Rodgers is well aware of himself - and a positive first season in charge for the Northern Irishman is a step in the right direction.
"Liverpool have high standards. That is what you must be judged by," Rush, who two European Cups, five league titles, three FA Cups and five league cups at Anfield, told Press Association Sport.
"Those standards were set by Bill Shankly in the 1970s and Bob Paisley took them on. It is a difficult level to attain but those are the standards.
"Why shouldn't that be the judgement? No one is saying he has to get there straight away, that he immediately must start winning the Premier League or the Champions League.
"But that has to be the aim. The key is to maintain improvement. If you can do that every year eventually you will get there."