The 22-year-old impressed under Steven Gerrard this season, providing six goals and six assists in the Scottish Premiership.
Klopp will evaluate in pre-season whether Kent will have a part to play in his first-team plans next season, according to a report in the Daily Record.
“Jurgen has heard a lot of good things about Ryan’s performances at Rangers and he has already made it clear he wants to take a good look at him during the club’s pre-season training,” a Liverpool source told the newspaper.
“Obviously Rangers would prefer to get another loan deal done as quickly as possible but it looks like they’ll have to bide their time on this one.
"It’s almost certain that no decision will be made until the end of July or the start of August when Jurgen names his squad for next season’s Premier League.”
The winger picked up the PFA Scotland Young Player of the Year award for his performances in Glasgow and Gerrard is keen to retain the services of the youngster.
The manager has already made it clear to the Anfield hierarchy that he would like to keep Kent at Ibrox for another year.
However, Aston Villa are among several clubs considering a bid for the £7 million-rated player, while the Gers could also raise funds for a permanent deal by selling top scorer Alfredo Morelos.
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