The centre-back was left out of Suthampton's squad for the 1-1 draw with Huddersfield Town at St Mary's on Saturday, intensifying speculation over his future at the club.
Manchester City have heavily been linked with a move for the player, though it is thought that Spanish giants Barcelona are also seriously connsidering a move.
When quizzed on the former Celtic man's absence following Saturday's game, Pellegrino insisted that there was nothing untoward in his decision, saying: "I decided that was best for this moment. This is the reason. I am responsible for picking 11 players, and I intend to pick the best for the team.
Though the Argentine didn't completely rule out a January departure for his star man.
"Around him will be a lot of speculation and we have to wait to see what happens," Pellegrino explained.
"Van Dijk's future? I don’t know, we’ll have to wait, but I will repeat, there will be a lot of speculation around him. We have to wait."
Could the Premier League and La Liga pacesetters be set to go head-to-head over a transfer?