Liverpool have plundered St Mary's over recent seasons, with Adam Lallana, Rickie Lambert, Dejan Lovren and Nathaniel Clyne all heading to Anfield.
And the London Evening Standard claim Liverpool are once again gearing up to sign another of Saints' best players.
According to the report, Liverpool are aware that Southampton will demand at least £50m for Van Dijk – the fee Manchester City paid Everton for John Stones last summer.
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And Reds manager Jurgen Klopp highly rates the centre-back.
Chelsea pushed hard to sign Van Dijk last summer but Southampton held firm on their reluctance to sell.
The Blues remain interested in the Netherlands international, who is supposedly open to leaving Southampton.
Liverpool and Chelsea want to sign Virgil van Dijk
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Van Dijk is reportedly worth over £50m
Chelsea tried and failed to sign Van Dijk, 25, last summer
In December, Saints boss Claude Puel denied the 25-year-old had a £25m release clause in his contract.
Liverpool have endured a mixed season at the back this term, with Mamadou Sakho's exile from the squad hampering Klopp's efforts to shore up the backline.
Meanwhile, Klopp insists he is not interested in signing a new goalkeeper to boost his defence, even though the form of both Simon Mignolet and Loris Karius have come under fire this season.