At this stage, Liverpool are reluctant to rule out a return for Van Dijk this season, saying such a determination can not be made until the player has his operation.
The club has yet to set a date for Van Dijk's surgery due to the necessity for the swelling around the knee joint to subside sufficiently for the player to be operated on.
But having replaced the out-of-form Joe Gomez with Joel Matip, playing his first match since the opening day of the season at Goodison Park, Klopp faces having to pair the untested and injury-troubled duo against Ajax in the Champions League on Wednesday.
Van Dijk has been a crucial presence at the heart of Liverpool's defence since arriving from Southampton and, until limping out of the Everton game, had played every minute of the 74 Premier League fixtures prior to the clash at Goodison Park - a run dating back to September 2018. "They are that outstanding".
"For Jurgen Klopp and the club, right away it's who can we get in?"
"I know the transfer window has just finished by they have to be ready on January 1 to bring someone in".
There are parallels with the serious knee injury that Manchester City centre-back Aymeric Laporte suffered at the start of last season.
The sidelining of Van Dijk is compounded by the absence of goalkeeper Alisson Becker with a shoulder injury.
That left the Reds three points behind their city rivals at the top of the table.
After the game, he headed for hospital on crutches for a scan on his knee and now the results have been revealed.
"Outrageous! I nearly want to say: I can't sleep on it".