The Reds have been linked with a move for Lazio centre-back Stefan de Vrij, who has also attracted interest from Inter Milan.
A £70m transfer would make Van Dijk the most expensive defender in football history.
Southampton reportedly believe that their valuation of Van Dijk is reasonable as he is eligible to play in the knockout stages of the Champions League and has more than four years left on his contract.
Speaking after Southampton's defeat to Chelsea over the weekend, Saints boss Mauricio Pellegrino said he did not know if Van Dijk would still be at the club when the January transfer window shuts.
"I can't control the market, I am not the owner," Pellegrino was quoted as saying by the Liverpool Echo.
"I have my opinion, I have to give to the technical director and the chairman my opinion, but the club will decide what's best.
"I can't control the market, and I am not the owner of the players," he reiterated.
"I will recommend to keep our best player here, at the club."
Liverpool pursued Van Dijk as a priority target early on during the summer transfer window but formally ended their interest after Southampton threatened to report the Reds to the Premier League over an alleged illegal approach for the player.
City manager Pep Guardiola is also interested in Van Dijk as he wants to strengthen his defence, while Arsenal and Chelsea are also reported to be keeping tabs on the Netherlands international.
Van Dijk joined Southampton from Celtic in September 2015. He signed a six-year contract extension just eight months into his career at St Mary's, with his deal running until 2022.