Saints' new boss has stressed the club still expect wantaway defender Van Dijk to feature this term, although he will miss Saturday's Premier League opener against Swansea due to a virus.
Dutch defender Van Dijk has submitted a transfer request amid persistent interest from Liverpool, but manager Pellegrino remains adamant Saints will not give up on the former Celtic star.
Van Dijk appears set on forcing a move however, and Pellegrino admitted there could come a point where Southampton may not be able to alter the 26-year-old's stance.
"My reaction is the same that I said three weeks ago: we are expecting him (Van Dijk) to come back, to train with us because we need the players, we need all of them," said the manager.
"Hopefully he can change his point of view. Virgil is an important player and we need him focused on our club."
Asked if Southampton can genuinely retain an unhappy player, Pellegrino said: "But the club was clear from the beginning with the player.
"We can't control every single situation. We can decide our dynamic in this group and Virgil is part of our group; he has a contract.
"He has spoken with (executive director) Les Reed. We've spoken with the player but not too much, a couple of meetings.
"I was clear, the club was clear, he was clear on his position. But we have to keep going because I can't control the thinking or the feeling of the player."
When quizzed on whether Van Dijk would be available for selection this weekend, Pellegrino added: "No, no. Because this week he was with some trouble with a virus and he couldn't train the last two days."
Pellegrino confirmed Saints will continue to chase more new recruits during the transfer window.
Club record signing Mario Lemina could make his competitive debut against Swansea at St Mary's on Saturday, with fellow new arrival Jan Bednarek also in the mix to feature.
Confirming his desire to make further signings, Pellegrino said: "Yes, there are certain positions where we would like to add strength. But we never know because the market is difficult. It's not easy for us.
"For me with Mario (Lemina) it's a mix. He's a defensive midfielder or a box-to-box midfielder and he has quality, strength in the middle. We didn't have this profile of player I think.
"Physically Mario's good, he's been training at Juventus.
"Jan (Bednarek) is a young player that I think has the capability to adapt to the Premier League. And Mario is a player that is coming from one of the biggest clubs in the world.
"We are happy with that. Both of them can help us to be stronger."