Van Dijk was left out of the Southampton squad for the entire month of August due to transfer speculation.
Liverpool's chase of Dutch defender Van Dijk was one of the transfer stories of last summer.
The Dutch worldwide is one of the best center halves in the league, he was linked with a move away from the Saints in the summer as well but eventually stayed.
The Argentine - who accused unnamed players of lacking "commitment" after the 4-1 midweek loss to Leicester - was unable to confirm whether Van Dijk will remain at the club through the January transfer window.
Van Dijk was benched by his Argentine manager for his side's 1-0 defeat to Chelsea yesterday.
"I can't control the market, and I am not the owner of the players, I will recommend to keep our best player here, at the club".
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"We have got a lot of players in our club that the other clubs wants to buy or on loan".
"In that department we have too few players and to handle all competitions we are in trouble, real trouble if this is the case".
"I can't control this". He had a bad (knee) injury, but I think he recovered very well.
"After, the board can decide all together which is the best for the team".
Chelsea manager Antonio Conte was coy about whether he was still keen on signing Van Dijk.
"I will repeat that I can't control the market right now. The club was clear in the beginning and I have the same opinion".
Speaking after the match, Pellegrino said: "Obviously Virgil is an important player".