The Reds apologised for making an unofficial approach for the £70 million-rated defender in the summer but haven't given up hope of rekindling the transfer.
On Thursday, Southampton manager Mauricio Pellegrino said he couldn't guarantee van Dijk will stay at the club, admitting "everybody has got a price."
And if van Dijk impresses on Saturday, it could convince the Liverpool hierarchy to stump up the cash for a January move.