Speaking after Saints' 1-0 defeat at Stamford Bridge, Pellegrino admitted Van Dijk could be on his way in next month's window.
"II can have my opinion and this I have give to the technical director, to the chairman, my opinion".
Van Dijk had started Southampton's 11 games, after returning from long-term injury in September, but was dropped to the bench for their match at Chelsea on Saturday after some poor performances recently.
Independent.ie understands that Liverpool are considering other defensive targets in the January transfer window, with Southampton in no rush to sell Van Dijk unless a club gets close to their lavish £70m valuation of their star centre-back.
The Saints still maintain the stance that van Dijk is not for sale, but those interested believe they would have listen if bids went above £50million.
"We just speak about [January transfer signings] with the club and Txiki because in the central defender position we don't have many players, especially in this situation", said City boss Pep Guardiola.
"You never know what can happen in the future", he said.
"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 told ESPN: "I think this will happen, but I think it's probably going to happen in the summer". After that, the board can decide what is best for the team.
"I can't control this". I will recommend to try to keep our best player here at the club.