Van Dijk, Joe Gomez (knee), Joel Matip (ankle) and Trent Alexander-Arnold (thigh) are all working on their fitness and took part in Liverpool’s pre-season training in Salzburg, Austria.
“All four look really good, I have to say,” Klopp told the club website.
“Virg and Joe will not have double sessions now for the first few days with the team … I don’t know exactly when we can make the next steps but it will happen in the next four weeks, for sure we will get a big step forward.
“Joel looks good, absolutely, he is the closest probably to normal training… Trent was a serious enough injury but now not a major one, I would say. He is close but in the moment, probably the next five or six days he has a special programme.”
Liverpool, who finished third last season, begin their new league campaign at promoted Norwich City on August 14.
Additional reporting by Reuters.