The Premier League champions said on Sunday that Van Dijk has damaged knee ligaments following a challenge by Everton goalkeeper Jordan Pickford during Saturday’s Merseyside derby at Goodison Park.
Van Dijk’s knee operation has yet to be scheduled and Klopp did not put a timescale on the centre back’s return.
"A few things are clear... he now will be out for a while," Klopp told Liverpool's website here. "I expected it pretty much on Saturday after the game immediately, especially when I saw the challenge back.
“These situations are absolutely rubbish. You don’t need it but you have to get over it and Virgil will get over it, 100%, and today is already the first day, if you want, of the recovery.
Klopp said midfielder Naby Keita could feature in Wednesday’s Champions League match at Ajax Amsterdam after he resumed training following a negative COVID-19 result.