The No.3 was forced off in the first half of Tuesday's Premier League victory at Aston Villa due to a muscular injury.
That issue will sideline Fabinho for the meeting with Chelsea at Wembley on the weekend, though the Reds boss hopes there is a possibility he can appear again in 2021-22.
Offering an update on Thursday, Klopp told Liverpoolfc.com: "There's a good chance that he will be back whenever – not for the weekend but whenever."