Liverpool were determined to sign Van Dijk from Southampton earlier this month with Jurgen Klopp targeting a title tilt next season.
However, the Reds were reported by Southampton after making an illegal approach for the player.
Liverpool were then consequently forced to publicly apologise and a move to Anfield for the Dutchman seemed dead in the water.
Chelsea, Manchester City and Everton have also been linked with a move for Van Dijk, who has been a big hit at Saint Mary's since arriving from Celtic in 2015.
Virgil van Dijk to Liverpool: 10 other failed Reds transfers Fri, June 9, 2017
Liverpool have withdrawn their interest in a £60million move for Virgil van Dijk, but which other players have the Reds failed to land after a done deal looked certain?
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Express Sport runs through some of Liverpool's biggest failed transfers
Southampton star Virgil van Dijk is determined to join Liverpool
However, The Sun claim the 26-year-old only wants to join Liverpool.
And he is also considering handing in a transfer request to ensure he gets his wish.
Despite angering Southampton, Liverpool are still keen on a deal with Klopp refusing to give up.
The German sees Van Dijk as the perfect player to rectify their struggling defence – who shipped a huge 42 Premier League goals last season.
Liverpool could sign Virgil van Dijk from Southampton
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Jurgen Klopp wants to transform Liverpool into title contenders
The report adds Liverpool would have to pay a club-record £60m to sign the Holland star.
And that would gazump the £38m they paid Roma for Mohamed Salah last month.
Should he make the move to Merseyside, the defender will become the latest player to swap Southampton for Liverpool.
Dejan Lovren, Rickie Lambert, Adam Lallana and Nathaniel Clyne have all done so in the past three years alone.