The club confirmed it has no timetable for when Van Dijk will return to the pitch.
BeIN Sports reported the central defender could miss seven to eight months after suffering a significant ACL injury.
Van Dijk was subbed off in the 11th minute after being on the receiving end of a tough challenge from Everton goalkeeper Jordan Pickford. Pickford went to make a play on the ball and instead slid into Van Dijk's right knee.
The tackle looked bad, and subsequent replays raised concern about a significant injury given how his knee bent.
Compounding matters, Pickford avoided any punishment despite the fact replay officials looked over the play to see whether Van Dijk was onside. Professional Game Match Officials Limited later said Pickford's tackle wasn't subject to video review.
Van Dijk might be the single most important player for Liverpool. From the moment he arrived from Southampton, the 29-year-old has been the domineering presence at the back the Reds lacked following the departure of Sami Hyypia in 2009 and retirement of Jamie Carragher in 2013.
Liverpool aren't lacking in talent all around the pitch, but their Champions League and Premier League triumphs probably don't happen without Van Dijk's contributions.
Van Dijk's injury raises further questions as to why Liverpool didn't sign a center back following Dejan Lovren's transfer to Zenit St. Petersburg.
The reigning PL champions couldn't have expected the Dutch defender to be out for this long, but injuries are an inevitability. Now, Joe Gomez and Joel Matip are the only natural center backs with senior-level experience. Fabinho will probably have to feature more regularly in the back four, and manager Jurgen Klopp may need to count on Sepp van den Berg more than he expected until the transfer window opens again in January.