Netherlands captain Van Dijk was taken off at halftime in a 3-1 defeat at Brentford on Monday as part of a triple-change by Klopp.
"Virgil was a surprise for us and a big blow. He didn't feel a lot," Klopp told reporters, speaking ahead of Liverpool's FA Cup third-round tie against Wolverhampton Wanderers on Saturday.
"The diagnosis was pretty harsh. We're talking about weeks, more than a month, but I hope it goes quickly. It's hard for him, but he has played an incredible amount of games in recent years."
Liverpool are sixth in the Premier League standings with 28 points from 17 games.