Van Dijk had played every minute of the 74 league games prior to being forced off the pitch on Saturday, a spell stretching back to his substitution in the 3-0 home win over Southampton in September 2018.
Van Dijk has been a commanding presence at the heart of Liverpool's defence throughout the team's rise to the top of English and European football in recent years.
Although Van Dijk was able to walk off the pitch following lengthy treatment, sources have told ESPN that Liverpool are concerned by the extent of the injury and are determined to get a clear diagnosis by sending the Netherlands worldwide to see a consultant on Sunday.
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.
"The centre-back damaged knee ligaments following an incident involving Blues goalkeeper Jordan Pickford after six minutes at Goodison Park".
"He plays on with pain, he never misses a game". A good game. I am not anxious about the boys.
"I felt at the start of the season that Liverpool were still the team to beat".
Liverpool have lodged a request with the Premier League asking for a review of the VAR decisions.