Jurgen Klopp looked to have missed out on Van Dijk as the Saints reported the Merseyside outfit for tapping up the 26-year-old centre-back – forcing an apology from the Reds.
However, the Sun claim Van Dijk could still move to Anfield, with the Dutchman ready to hand in a transfer request at St Mary’s ahead of the 2017/18 campaign.
It’s understood he only wants to play for Liverpool next term despite Premier League rivals Manchester City also keen on a deal.
Southampton are demanding a fee of at least £60million for the Netherlands international this summer so a move will not come cheap for the Reds.
Liverpool have signed Adam Lallana, Rickie Lambert, Nathaniel Clyne and Dejan Lovren from Southampton in recent seasons.
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Chelsea are also believed to have expressed an interest in Van Dijk in recent weeks.