Chelsea remain in the hunt for the wantaway Saints centre-half, with Manchester City and Liverpool also keen.
However, Jurgen Klopp's side face a struggle to land their top defensive target.
According to Yahoo Sport, Southampton will not be bullied into offloading Van Dijk to Liverpool.
And as a result, the 26-year-old will more likely be sold to either Chelsea or City should he leave this summer.
Van Dijk is continuing to train away from Mauricio Pellegrino's first-team squad.
But Southampton are still hopeful the Dutchman will stay.
And club officials are apparently convinced they can get Van Dijk to commit to the cause once the transfer window shuts.
Southampton are yet to receive any offer near their valuation.
Yahoo Sport adds, though, a bid in excess of £70m may tempt Southampton to complete a deal.
Should Chelsea be willing to break their transfer record to bolster their backline, Antonio Conte could land his man.
Van Dijk wants to join Liverpool, but Chelsea and City may receive encouragement if they are ready to fork out big money.
Reports have suggested a £50m bid will be made by the Blues this week.
Van Dijk hasn't played since injuring his foot against Leicester in January.
