Liverpool were favourites to sign Van Dijk in the summer transfer window but made a public apology after being accused of making an illegal approach for the player.
As the two clubs prepare to meet in the Premier League this weekend, Pellegrino said Van Dijk's future could be decided in January.
"I can't control the market," the Argentine manager told reporters on Thursday when asked about Van Dijk's future. "You never know, because when we talk about money everybody has got a price.
"The budget is to sign players and the biggest teams have got money to try to sign players.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"I can't decide about that. This is a question for our owners... We have got a lot of good players and most of the teams want to sign them."
Southampton will be without midfielder Mario Lemina, who has missed the Saints' last two games due to an ankle injury, for the trip to Anfield and Pellegrino said the 24-year-old still has a way to go before returning to first-team action.
"Mario Lemina was training well, much better than before, but I think he will not be ready to play on Saturday," Pellegrino said.
"He has improved a lot but still already it is too soon to put him again on the pitch."
(Reporting by Aditi Prakash in Bengaluru; Editing by Toby Davis)
