The defender hasn't featured for the Reds since coming on as a substitute in the first pre-season friendly of the summer against Tranmere Rovers at Prenton Park on July 12.
Clyne is continuing to work with the club's medical team at Melwood in order to return to full fitness as soon as possible, though he was not named in Liverpool's Champions League squad for the group stages of the tournament due to the issue.
Should the Reds progress in the competition, the 26-year-old will be eligible to be registered for action in Europe in the New Year.
Klopp told Liverpoolfc.com: "Nathaniel is working hard on his rehab and we have kept a close eye on his progress, but the reality is we are looking at some time still before he is back and available for selection. I am told it will be longer than just a couple of weeks, so we will continue to monitor and assess.
"It is pretty certain he'll not be available for the majority of the Champions League group matches. If we are fortunate enough to still be in UEFA competition in the New Year he could be added to the squad then.
"The medical guys are pleased with how he is responding and Nathaniel has shown a really strong attitude, given he is not someone who is used to being out injured.
"What is positive for us is how Joe [Gomez] and Trent [Alexander-Arnold] have stepped up and grasped their opportunities."