The Saints reported Liverpool to Premier League for tapping up their player earlier this week, which promoted the Reds to apologise to the Saints and end their interest in van Dijk.
However, The Mirror claims that despite accepting a public apology from the Reds, the Saints want the Premier League to conclude their investigation.
The statement on Reds website read: "Liverpool Football Club would like to put on record our regret over recent media speculation regarding Southampton Football Club and player transfers between the two clubs," read a statement.
"We apologise to the owner, board of directors and fans of Southampton for any misunderstanding regarding Virgil van Dijk. We respect Southampton's position and can confirm we have ended any interest in the player."