Rush, Liverpool’s all-time leading scorer with 346 goals during his two spells at Anfield, still thinks the Reds can finish in the top four of the Barclays Premier League.
Klopp’s team are currently ninth in the table, seven points behind fourth-placed Manchester City but with a game in hand.
Another route into the Champions League for Liverpool is via the Europa League where they face Klopp’s previous club Borussia Dortmund in the quarter-finals next month.
Rush has been impressed with Klopp since he became manager in October but he expects to really see the fruits of the German’s work next season.
“I think he has done very well so far,” Rush, speaking at the Cardiff leg of Carlsberg’s UEFA EURO 2016 trophy tour, told Press Association Sport.
“But I think next season is when we will really start to see his work, when he has had a full pre-season with the players and another transfer window. I think then we will see Liverpool really kick on.
“There will be players he will want to move out and players he will want to bring in, and I think Liverpool will look to strengthen in every area of the squad.
“But this season is not over yet for Liverpool. I still think the top four must be the aim – and I think it is achievable. I said it at the start of the season and I still believe that now.
“We are still in the Europa League, it is a difficult draw but I think the winners of that tie could go on and win it. So there is still a lot to play for this season.”
Klopp came close to winning a trophy last month as Liverpool lost to Manchester City in a penalty shoot-out in the Capital One Cup final at Wembley. Rush thinks it could be a sign of things to come and that the Merseysiders will be capable of regularly challenging for silverware under Klopp.
“It is a shame we couldn’t go on and win the League Cup but to win trophies you sometimes need that little bit of luck and we didn’t have it that day,” said the 54-year-old former Wales international.
“But there are good signs there and in time [Klopp] will assemble a squad, not just a team, a squad capable of competing with the very best – because that is what Liverpool should be doing.
“I think he is beginning to realise the expectations among Liverpool fans but I think he has the fans on his side with the enthusiasm and passion he shows on the touchline.
“Hopefully it will all start to really come together next season.”
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