Liverpool struggled in the absence of Van Dijk after the centre-back suffering a season-ending knee injury in the Merseyside derby at Everton in October last year.
Although Klopp also revealed fellow defender Joe Gomez is closing in on a return to action, with the 24-year-old having suffered his own serious knee injury while on international duty with England last November.
Speaking to the official club website, Klopp said: “I hope, I am not sure, that there is the opportunity that Virgil could play a few minutes. He looks really good in training and maybe we can bring [him in], but I need to have some final conversations. He looks ready and we will see.
“If you see the games now as part of their rehab training, that makes sense. Joey is very close. There is no race between the two of them; they had different injuries and stuff like this, but [he is] very close.
“If Virgil can play now 20 then probably Joey can in the game after. We will see. In training they both look really good.”
Meanwhile, the return of some key defensive reinforcements will be a major boost for a Liverpool side who will be looking to regain the title from Manchester City.
However, Klopp is keen to not rush Van Dijk and Gomez back into the fold prematurely, hinting he is prepared to ease them back into match action. Liverpool kick-off their Premier League campaign on August 14 against newly-promoted Norwich at Carrow Road, live on Sky Sports.
“I think from time to time there are some pictures of them from training sessions and everyone can see how good it looks, but there is no rush for us,” Klopp added.
“It is not about they now need five pre-season games to be ready for the start or whatever. It is a long season and we prepare them after these very, very serious injuries for the rest of their careers and not for the first game of the season.”