32-year-old Thiago has not played a single minute of football in 2023/24, following his recovery from hip surgery at the end of last season.
The former Bayern Munich star, in the final year of his Liverpool contract, had been predicted to return in pre-season, but has seen his comeback delayed.
Back in October, Klopp revealed that 'annoying setbacks' in the player's rehab had stopped him taking part in team training and being available for selection.
He said: "It’s just, for him (Thiago) especially, it’s annoying, he's now had two setbacks in the rehab.
"Not massive but enough to take him off the pitch again. Obviously not the same injury, not at all, but a bit similar."
Meanwhile, assistant manager Pep Lijnders admitted that Liverpool 'have missed' Thiago's presence in midfield.
While most official updates have been coy about the midfielder's recovery timeline, Klopp has now put a date on his number six's possible return.
Speaking in his pre-match press conference ahead of the trip to Luton Town on Sunday, Klopp revealed: “With Thiago, it’s an ongoing thing and we cannot put any kind of pressure on there as well.
“We expect him to be back - and I would say probably the start of the New Year. It’s now already November, so that would mean another four weeks. I would say that’s probably a realistic thing to try and I can’t wait, to be honest, to have him back.
“The player he is is just exceptional. I understand you ask, but it’s nothing where we can have a week by week update because that will not change a lot in this moment.”
Meanwhile, Klopp was not able to provide much of an update on the Reds' other injured midfielder, Stefan Bajcetic – who has suffered similar difficulties after a long-term issue.
"With Stefan, it's one of these things where Liverpool have a brilliant player," Klopp said.
"His quality, potential, everybody can see. What a player. But his body was not ready yet. We have to put the brake on and wait until it's all fine. Then Liverpool has a wonderful player to use."