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The Reds secured a 19th league triumph with a record seven matches to spare when Manchester City lost 2-1 at Chelsea on Thursday.
Rush said the current squad deserved comparison with the all-conquering side of the 1980s, in which he played, but felt Jurgen Klopp's side showed a superior work ethic.
"When we won the league we would go out, celebrate and have a great night," Rush told Britain's Press Association.
"We won once with five or six games to go and after that I'm not sure if we won a game," added the former striker ahead of eSoccer Aid for Unicef, taking place on Saturday.
"In a season your job is to win the league and when we won it the training was relaxed."
But Rush said Klopp's players showed an impressive hunger.
"Times have changed and these players now are so focused and fit," he said. "If you look at this lot now they won't stop, they'll keep going because they want to win every game.
"They'll be saying 'We won the league, let's go for City's 100 points (a record set in 2017/18)'."
Rush, who scored 346 goals for Liverpool, played in the last Anfield side to win the title, in 1990.
The former striker said it was the job of Klopp's team "to set records" for Liverpool.
"To break any Liverpool record is an achievement in itself," he added. "These can break more records so they have to be up there with the best.
"The average age of the team is also very good so over the next few years hopefully Liverpool will be quite successful."
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