Klopp told the Liverpool website: "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."