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 quick. It's hard for him, but he has played an incredible amount of games in recent years.
"It's the hamstring. He never had an issue with muscles before, never mentioned anything. It just now happened, it was one sprint too much."
Van Dijk's last lengthy layoff came in the 2020-21 season, during a period in which Liverpool's defensive resources were depleted by injuries to Joe Gomez and Joel Matip as well as stand-ins Fabinho and Jordan Henderson.
Klopp added that Henderson, who missed Liverpool's defeat at Brentford due to a concussion, was back in training, saying, "he did exactly what he was allowed to do yesterday, training without heading, and from today he is allowed to do everything."
The German manager also heaped praise on Liverpool's new signing Cody Gakpo, who is in line to make his debut against Wolves on Saturday after joining from PSV Eindhoven, backing the Dutch forward to quickly adapt to life in England.
"He's a good footballer, good in small spaces, likes to shoot from distance, good finisher and dribbler," Klopp said.
"Everybody who wants to know about him could know as there are a lot of videos out there about him. It is nice to see in real life, in front of you.
"He looks really promising and fit. The sessions are intense for him because they're different. I'm very positive about him. Coming from a new league is always difficult, but we expect a positive impact."
Prior to his move, Gakpo had been linked with other Premier League clubs, including rivals Manchester United, but Klopp said the 23-year-old's decision to come to Anfield despite Liverpool's position in the table spoke to his character.
Liverpool are currently sixth with 28 points from 17 games, seven points off the top four, and have a tough challenge to secure Champions League qualification for next season.
"There have been easier moments to sign for Liverpool," Klopp said.
"This year we can't guarantee Champions League football, but Cody never asked. He knew the decision. He can see the table. He's a guy who doesn't want to jump on a running train, he wants to push the train, that always helps."
