Van Dijk has told Liverpool he wants to join them, raising the possibility of a stunning £50million deal.
The Dutchman has established himself as one of the Premier League's best defenders since moving to English football from Celtic in 2015.
Southampton are determined to keep hold of the 25-year-old but could be forced to sell if the player pushes for a move.
Should Van Dijk arrive at Anfield, his defensive partner could be Lovren but it appears the Croatian was unwilling to discuss his potential new team-mate.
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I’ll tell you again, we do not need to sell anybody
Southampton chairman Ralph Krueger
While in Sydney on Liverpool's post-season trip to Australia, Lovren was asked about the rumours surrounding Van Dijk.
But he stormed away from reporters saying: "I don't want to talk about that!"
If Van Dijk leaves the south coast, he would become the sixth player to quit Saints for Liverpool since 2014.
Adam Lallana, Dejan Lovren, Sadio Mane, Nathaniel Clyne and Rickie Lambert have all moved to Merseyside in the past three years costing nearly £100m.
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Virgil van Dijk appears to be edging closer to a move to Liverpool
Last month, when asked if Southampton could reject bids as much as £60m for their best players, chairman Ralph Krueger said: “Can we? Yes.
“The big thing is here again I want to underline the strategy, less the detail.
“You know I really respect our lead in football, Les Reed is doing an excellent job, and Ross Wilson has taken on a big role assisting Les and leading our football-specific decisions.
“They will continue next week once they’ve analysed the season and really de-brief completely, they will make then the football decision we need to make.
“I’ll tell you again, we do not need to sell anybody.”