Van Dijk has again been linked with Liverpool and Chelsea when the transfer window opens next month, and Pellegrino refused to rule a move out for the Dutchman.
The Gunners are not alone in the race for the Netherlands global as Barcelona, Manchester City, Liverpool and Chelsea are keen as well.
Van Dijk, who has made 12 appearances for Southampton this season, was dropped by Pellegrino for Saturday's 1-0 defeat to Chelsea.
"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".
He said: 'As you know very well, I don't like to talk about players of the other team.
He added: "I can't control this". I will recommend to try to keep our best player here at the club. Virgil Van Dijk is clearly a hugely talented player who would improve nearly any team on the planet but £70 million is a lot of money for a centre-back.
"He came in for the first training session and he was that good defensively, I can remember Neil Lennon looking at Garry Parker and saying, "He could play centre midfield". He had a bad injury, but I think he recovered very well. "The club was clear in the beginning and I have the same opinion".