"The club should be prepared to fight because it is really important that we continue to be regarded as the most successful club in the country," said Rush in his Liverpool Echo column.
"That record has been built up over decades of hard work and doing things the right way and it has to be preserved.
"No-one has the divine right to win the title. You have to earn that right and that's what Liverpool have done in the past and need to do again.
"This time, United have been pushed all the way by Chelsea. It was a tight finish and could have gone either way. But United did enough on the last day and that makes them worthy champions.
"From a Liverpool point of view, we have to be on their shoulders next season and we have to be involved in the title race as it goes into the final straight.
"That is the challenge for everyone at the club and what greater motivation do you need than preserving one of the proudest records in football?"