The Dutch international has been sidelined with a foot injury since January and is unlikely to feature against this campaign. Jurgen Klopp has set his sights on strengthening his side's back four for the next season.
According to the Liverpool Echo, the Reds are keen on bringing Van Dijk to Anfield after the end of the season. The centre-back joined the Saints from Celtic in 2015 and the Merseyside club's former manager Brendan Rodgers wanted to secure his services.
However, Liverpool's scouts advised the Northern Irishman not to complete Van Dijk's signing. He joined Southampton and has developed into one of the best defenders in the Premier League under Claude Puel's guidance this term.
Van Dijk's form prior to his injury has seen him attract interest from the top clubs in England. Apart from Liverpool, league leaders Chelsea, Pep Guardiola's Manchester City and Everton are in the race for his signature.
The same report claims Arsenal have also shown interest in signing the 25-year-old in the upcoming window. Southampton are demanding a fee of £50m ($64.3m) for their captain and Liverpool are not put off by the hefty price tag.
The Reds' previous club-record was £35m ($45.1m) that they paid to Newcastle United in signing Andy Carroll in January 2011. However, Liverpool's hope of signing Van Dijk hinges on their chances of qualifying for next season's Champions League.
Klopp's men are third in the table with 66 points from 34 games. They are two points ahead of fourth place Manchester City and three ahead of Manchester United, who are fifth in the table. However, the two Manchester clubs have the advantage of having two games in hand.
Meanwhile, Puel has expressed his desire to retain his star player beyond this season. It should be seen whether Southampton can retain Van Dijk, especially if any of the clubs meet their asking price.
"I cannot prevent all the newspapers from speculating about him. It's always the same thing, [Van Dijk] has a long contract with Southampton, he's our captain, he's very important for us and I want him for next season," Puel explained.
"He cannot play until the end of the season but now he works normally and recovers. He will be available for training soon.
"I hope we can keep all our good players, it's important to continue the work with individual players, it will be important to keep them and also to took take information about different players. This is not just to speak about Virgil, he's a Southampton player and our captain."