The Frenchman came off with a blow to the knee in Liverpool's final pre-season game - a 3-0 defeat to Strasbourg - and will not be available for the start of the Premier League this weekend.
"Unfortunately, I have to spend some time off the pitch but it won't be too long. In the meantime, I will be doing my best to come back even stronger and supporting the boys," Konate wrote on social media.
The length of Konate's absence is still unspecified, but Klopp is now left with Joel Matip, Virgil van Dijk, Joe Gomez and Nat Phillips as his centre-back options for Saturday's Premier League opener with Fulham.
Phillips has been strongly linked with a move away from Anfield after falling back down the pecking order over the last 12 months, with Bournemouth and Fulham both keen on the 25-year-old.
However, with Konate now sidelined, Klopp has hinted that he will not sanction a move away for Phillips in the current transfer window after already allowing Rhys Williams to join Blackpool on loan.
"Yeah, probably. It will influence the transfers – outgoing transfers – I would say," Klopp told reporters at Friday's pre-game press conference when asked what Konate's injury would mean for Phillips's future.
Konate is one of several injury concerns Klopp must contend with for the opening day of the Premier League season, with Diogo Jota, Caoimhin Kelleher and Alex Oxlade-Chamberlain already confirmed absentees.
Alisson Becker will be back in goal after recovering from an abdominal issue, but Klopp did not have good news to share on the conditions of Naby Keita, Curtis Jones and Kostas Tsimikas, all of whom could miss out.
"Ali [is] fine. We hope with Curtis we caught it in the right moment, that's just how it is with young players," Klopp added. "They have some kind of stress reactions sometimes and then if they go over it because it's not massive pain, it could lead to something serious.
"We caught it early but we still have to be careful. That's it with him, so he is not in. Naby is ill, but I think he will be back today the first day. I underestimated it a little bit when I said he will definitely be fine.
"He came in yesterday and still showed some little symptoms, no COVID but it is not right. I guess he will be back today.
"Then Kostas [was] unlucky in a training situation. I saw him now, he was out on the pitch and he might be - in his opinion, anyway - ready to train next week, which is helpful.
"Ibou, in a challenge in the game against Strasbourg, what happened there we have to see how long it will take but he is out for a while. Oxlade [is also out], of course."
"We have too many injuries, that's true, but in the moment nothing changed. There were a lot of unlucky situations; started with Caoimhin, started with Diogo pretty much. One was injured, the other one got a re-injury, it was not 100 per cent clear.
Liverpool enter the new Premier League season having already seen off Manchester City 3-1 in the Community Shield, and Klopp must also decide whether to start Darwin Nunez or Roberto Firmino at the tip of the attack at Craven Cottage.