Southampton have insisted that Van Dijk is not for sale but the Dutchman could continue to sit out of action in a bid to force through a move.
Chelsea boss Antonio Conte has been a long-term admirer of Van Dijk and has finally convinced Stamford Bridge chiefs to submit a bid.
Conte is hopeful that he can still secure up to three more additions before the transfer window closes at the end of the month.
Liverpool had been leading the race, but Chelsea are hopeful that they can land Van Dijk before the big Premier League kick-off next weekend.
Southampton will likely have to give Liverpool the green light for them to make a move given the previously poor relations between the two clubs.
Meanwhile, Pellegrino is determined to convince the 26-year-old to stay with the Saints.
He said: “I would like to bring the boy and to persuade him that he’s important for us and the club will not sell him.
“We are expecting the best for everyone, even for Virgil, but he doesn’t want to be part of the team.
“This is the reason he is aside from the team.”
Van Dijk is going to be short of match fitness having not played since suffering a season-ending ankle injury in January.
And despite Van Dijk’s protests, Pellegrino insists the Saints’ position has yet to change.
He added: “We continue in the same way. The situation is the same.
“I can’t say no more, because the situation is the boy is aside of the team because he doesn’t want to play for us.”