Van Dijk appeared just five times for the Reds in the Premier League last season and missed the Netherlands' Euro 2020 campaign due to an anterior cruciate ligament injury sustained against Everton in October.
The 30-year-old was included as part of Liverpool's squad for its warm-weather training camp in Austria this month, but he was not used in last week's 1-0 win over Mainz and Klopp made clear he would not rush the defender back.
However, the Liverpool boss is now ready to give Van Dijk some minutes at Tivoli Stadion, while fellow centre-back Joe Gomez – who has not featured since injuring his knee while on England duty last November – is also closing in on a return.
"I hope, I am not sure, there is the opportunity 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'll see," Klopp said.
"If you see the games now as part of their rehab training, that makes sense. Joe 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.
"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.
"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.
"Losing a little bit of patience for them to return as a supporter or as a coach even is not bad, it just means they were pretty good and we want them to be back in the team."
The imminent return of Van Dijk is a major boost for Liverpool, who sorely missed the Dutchman's leadership and defensive attributes last season, having played every match in their successful 2019-20 Premier League campaign.
Only one defender in the league bettered his 191 aerial wins during that season, while he attempted and completed more passes than any of them. Trent Alexander-Arnold, meanwhile, was the sole defensive player to have more than his 3,624 touches.
Liverpool won 81.4 per cent of the 43 Premier League matches Van Dijk played between August 2019 and his injury, but that fell to just 51.5 per cent between October and the end of the season when he was out of the team.
Van Dijk is not the only player in line to return to the Liverpool side this week, with Andy Robertson and Diogo Jota set to feature against Bundesliga side Hertha.
Robertson and Jota represented Scotland and Portugal respectively at Euro 2020, but both players linked up with their club colleagues last week and are now deemed ready to feature in pre-season action.
"They will have minutes, that's the plan. They obviously trained completely normal all the time now and we try to involve them," Klopp said.
Liverpool also have friendlies against Athletic Bilbao and Osasuna to come before beginning their Premier League campaign against Norwich City on August 14, so Klopp will continue to rotate his side during the upcoming matches.
"Some of the players are ready for 60 minutes. It will be hard for them definitely, but it means then others play 30 minutes and stuff like this, or even maybe 20 minutes and these kind of things," he said.
"Pre-season is about trying. It's a proper, proper challenge to make all these individuals to bring them in the best possible physical shape and make us more ready or stronger as a team.
"Unfortunately, football is not like cycling, so you don't miss it completely but the pedals and stuff like this just go away so quick that's incredible. So we have to work on that again, that's what we do.
"Today is a really tough [double session]. I like it, the players probably sleep now already and that would be better because this afternoon it would be really hard."