England winger Downing arrived from Aston Villa last week for a reported 20 million pounds, joining Jordan Henderson and Charlie Adam as Liverpool's close season signings.
Rush, Liverpool's all-time top scorer, told reporters in Singapore on Sunday that Downing would get the best out of Andy Carroll, the former Newcastle United striker who joined Liverpool in January for 35 million pounds.
"Andy [Carroll] likes the crosses coming in to him. So, hopefully, Downing can supply the right balls for him," said Rush on the sidelines of a coaching clinic.
"Charlie can score goals from anywhere and can pass the ball and I believe Jordan will be an England regular one day," added the 49-year-old.
Liverpool finished sixth in the English Premier League last season, missing out on a European spot, and Rush believes the key to winning the league is getting the new faces to settle in quickly.
"You're looking at six teams who could win it. But whoever's new signings settle in best and early on will be the side that win the league," added Rush, who was a member of the last Liverpool team to win the league in 1990.
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