The 22-year-old spent last season on loan at Ibrox, his fifth temporary spell away from Anfield.
He became a fans' favourite with a number of exciting displays, helping Rangers to a second place finish.
It was hoped that with Gerrard's connections and Kent's success in Govan the club would be able to land him on a loan deal once more.
However, Klopp said: "Ryan has improved a hell of a lot. He came back a completely different player.
“It was a very important year with a good manager at a really good club. It was exactly the right thing at the right moment for him.
“If Rangers can buy him [he can go back]. The time is over for him to go on loan.
“You put players on loan until a specific age but they need to find a place where they can settle.”
Now, it appears that Leeds United could be set to swoop.
He was part of Liverpool's pre-season plans, making five appearances and even catching the eye of fans.
One fan noted that Kent was "encouraging" but "often seems to look good in pre season... before he's inevitably shipped off somewhere."
Another said: "He's an interesting player, I'm not quite sure whats missing from his game, he always looks a threat and a nightmare to defend against, very two footed and strikes a ball well."
During 3-2 defeat to Borussia Dortmund in the US, Kent played a vital part in one of the goals with a "brilliant dummy".
It prompted one supporter to view him as the best of the young talent trying to make an impression.
They wrote: "So far I think Kent has been the most impressive of the young players. Just steady and calm in and out of possession. Seems to make fewer mistakes than the other youngsters."
However, it wasn't all positive.
"Don't think there is a chance for Kent here," said on Red.
Another stated: "He may be able to have a PL career if things pan out perfectly for him, but he just is not good enough to make it here and never will be. Most players aren’t to be fair to him. Needs to move on."
There was a similar message on the popular Liverpool podcast The Anfield Wrap as they looked back at pre-season.
One pundit said: "I think this pre-season, this tour, it was the opportunity for the players like Wilson and Kent to absolutely stamp their authority on the team in the absence of the other front players.
"It was an opportunity for them to show what they are about and make the most of that opportunity, and I just don't think that they did.
Another added: "It's not just that they have been put in the shop window but they have got a full pre-season under their belt with one of the best managers in the game at giving people a pre-season, which is a real selling point."
Gerrard has noted that Rangers may have to move on from the player but he will be patience with more than 25 days of the transfer window remaining.
Although a complication could arrive with Leeds United forward Kemar Roofe on the verge of signing for Anderlecht in the Belgium.
Kent is the key target to replace the striker..
Leeds United reporter Phil Hay said: "Should Roofe to Anderlecht get done, highly likely that Leeds will try again with Liverpool’s Ryan Kent. A player Bielsa has wanted all along."