Konate picked up the issue in the dying seconds of the Reds' 2-0 Champions League win over Real Madrid after a clash with Brazilian striker Endrick.
The French international's form has been key to Liverpool's stunning start to Arne Slot's time in charge.
Liverpool are eight points clear at the top of the Premier League and have won all five of their Champions League games.
Konate has started every Premier League and Champions League match since replacing Jarell Quansah at half-time of the season opener at Ipswich.
Liverpool have not disclosed how long the 25-year-old will be sidelined, but reports suggest he could miss up to six weeks.
"So frustrating to pick up this injury right at the end of a great game on Wednesday night," Konate wrote on Instagram.
"Now we start the recovery process but one thing I promise is I will come back and be the best I can be once again. Thank you for the amazing support at Anfield."
England international Joe Gomez is expected to deputise for Konate ahead of Quansah.
