The defender, who has not started a game since January, was a late substitute at Crystal Palace last week and made a goalscoring appearance for the Under-23s in a 3-0 win against Stoke City on Monday.
Van Dijk attempted to force a move away amid sustained interest from Liverpool over the summer, only for the Saints board to reject all advances and keep hold of their captain, who was temporarily frozen out of the squad.
Ahead of Southampton's Premier League meeting with Manchester United at the weekend, Pellegrino told reporters: "We will decide on Friday, but every week for Virgil is important because he is getting better.
"After a long period of time without playing a game now he is training well, and it's good news for us because we need Virgil like every single player in our squad.
"Virgil is an important player for us. I think every single week he will improve, but at the same time he needs to play, he needs to train. The best training session for me is to play games. Now he is playing in the reserves and he is training well."
Van Dijk has made 69 appearances for the south coast club since his 2015 move from Celtic.