The Holland captain has been out since undergoing surgery on the anterior cruciate ligament in his right leg after a clash with Jordan Pickford in October's Merseyside derby.
Van Dijk has sat out the first three training games at the club's camp in Austria but is in line for a short cameo against Hertha Berlin on Thursday.
"I hope, I am not sure, that there is the opportunity that Virgil could play a few minutes," manager Jurgen Klopp told liverpoolfc.com.
"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."
Fellow centre-back Joe Gomez, who has been out almost as long also with a knee injury, is likely to make his comeback in the Anfield friendly against Athletic Bilbao on August 8.
"If you see the games now as part of their rehab training, that makes sense," added Klopp.
"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 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 serious injuries for the rest of their careers and not for the first game of the season."
Left-back Andy Robertson and forward Diogo Jota will make their first appearances of pre-season against Hertha after joining the camp late due to their Euro 2020 involvement.
A number of players have yet to return after extended international commitments but one – Xherdan Shaqiri – may not come back.
The Switzerland international was one of the players Klopp was prepared to let go this summer and his performances at the European Championship increased interest with Sevilla and Villarreal now talking with the club about a permanent deal.
The club are looking for around £15million as they hold the option of an additional 12-month extension on the 29-year-old's current deal so he is effectively contracted until 2023.
"I told the Liverpool board I feel ready for a new challenge. They accepted my decision and will now consider bids to sell me. Liverpool won't stop me this summer," Shaqiri told Italian newspaper Corriere dello Sport.
"The important thing at this moment in my career is to be able to play regularly, but that hasn't always been the case in the last three seasons."
Liverpool are continuing to focus on outgoings – having already banked upwards of £27million from the sale of fringe players – and new contracts for a number of their existing players.
The PA news agency understands it is unlikely there will be further additions to the squad following the £36million arrival of centre-back Ibrahima Konate from RB Leipzig.
But they continue to strengthen from within with 18-year-old Conor Bradley, currently out with the first team in Austria, signing a new long-term contract.
Bradley, signed from Dungannon United in 2019, made his senior Northern Ireland debut in May and is viewed as potential cover for Trent Alexander-Arnold at right-back should Wales international Neco Williams depart this summer.