The Spain international has not played since last month's Merseyside derby when he was injured in a tackle which saw Everton forward Richarlison sent off.
While he has trained on a couple of occasions, the new signing is not currently ready to join in sessions and the PA news agency understands it will be early January before he is fit to play.
However, there has been no setback and no surgery is required but it has meant Thiago has played just 45 minutes at Chelsea and 90 minutes against Everton.
"Thiago we have to maybe clarify it a little bit," said manager Jurgen Klopp.
"On that day when Thiago got injured in that challenge in the Everton game, there was another bad injury (to Virgil van Dijk) and after the scans it was like one was really bad news and the other one was really good news because nothing was broken, nothing was ruptured and stuff like this.
"But, as we realise now, it was still a massive impact on the knee. The problem is in a situation like that, nothing is ruptured and nothing is broken and that means everybody thinks, 'OK, you can go after a few days', but the impact on the leg was that big that it is still not OK.
"It is not massive and he trains from time to time, not in the moment because we just have to realise that we have to go a few further steps.
"I can't say exactly when he will be fine, but it will take a few weeks still with Thiago."
Klopp will again have to juggle his limited resources – midfielder Naby Keita has a chance of being fit but Trent Alexander-Arnold is still a little way from being ready – for the Champions League visit of Ajax.
A draw will be enough for Liverpool to qualify for the knockout stage, by virtue of a better head-to-head against both the Dutch side and Atalanta, and mean next month's match against Midtjylland will be of less significance and Klopp could rest players.
"I actually don't see it in that perspective. That would mean I could count on the points already," said Klopp.
"We deal with the situation how it is. Three points would be great, absolutely great, and it is so clear it would help us massively but I didn't think I'd have to mention it.
"It is an important game against a really good team, that's the way we want to approach it."
The club will this week submit a planning application to Liverpool City Council for the proposed expansion of its Anfield Road Stand.
Plans for the submission – to increase capacity by 7,000 to over 61,000 – were paused in spring this year due to the uncertainty surrounding the COVID-19 pandemic.
As part of the application, the club is also seeking permanent permission to hold concerts and major events at the Stadium.