The defender recently handed in a transfer request in a bid to push through a move away from St Mary's, with the Reds, Manchester City and Chelsea all said to be interested, per BBC Sport. But new Saints boss Pellegrino doesn't appear to have given up hope of convincing the 26-year-old to stay.
"Virgil is an important player and we need him focused on our club," he said, per the aforementioned BBC Sport report. "My reaction is the same that I said three weeks ago: We are expecting him to come back, to train with us because we need the players, we need all of them. Hopefully, he can change his point of view."
As noted by Jeremy Wilson of the Daily Telegraph, the Dutchman trained alone in pre-season. Earlier in the window, Liverpool apologised to Saints for their pursuit of the player and said they would not be making another move for him.
Yet rumours have swirled, with the Reds continually linked to the star centre-back. The Empire of the Kop Twitter account believes getting Van Dijk through the door will be decisive in Liverpool's season:
Looking at Liverpool's squad, another body at centre-back would be beneficial. Van Dijk is commanding, physical and composed in his duties; while Dejan Lovren and Joel Matip made a solid pairing last season at times, the Southampton star would be a big upgrade.
After years of cashing in on their star men, often to Liverpool, it appears as though Saints are ready to stand firm in the face of serious interest. It's going to be fascinating to see whether that's a stance they persist with between now and the end of the transfer window.
Emre Can Not for Sale
As noted in the report, an earlier piece from the Italian newspaper had suggested Juve were hopeful of securing Can, with the midfielder turning down a new contract at Anfield. However, in the most recent edition, it's said the Bianconeri have been warned off chasing Can.
Liverpool are said to be keen to keep the rest of their squad intact as they get their Premier League campaign under way against Watford on Saturday.
Can has less than a year left to run on his current terms with the Reds, and some fans will be concerned a new contract has yet to be agreed, especially if a club the calibre of Juventus is sniffing around.
Karl Matchett believes it's vital Liverpool get him tied down:
The Germany international does have a lot to offer in the middle of the park for the Reds. Late last season, in particular, he was key in the team finishing in a UEFA Champions League spot. Can is authoritative, can dribble with the ball and is resourceful in possession when he bounds forward.
Given Juventus have long dominated the Italian top flight and have challenged consistently for Champions League titles, it'd be a tempting move for Can. Liverpool fans will want to see him tied down as quickly as possible, though.