Liverpool were forced to publicly withdraw their interest in the Southampton defender after being accused of tapping up the Dutchman by the south-coast club.
The Reds are said to still be keen on signing the 26-year-old though, with The Telegraph claiming they are willing to spend pay whatever it takes to land him before the window shuts.
But Southampton are refusing to sell their star man, meaning Liverpool may have to wait until January or next summer if the Saints continue to dig in.
Liverpool today announced the signing of RB Leipzig midfielder Naby Keita, who will arrive at Anfield next summer.
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Liverpool to spend whatever it takes to sign Virgil van Dijk
Liverpool to pay whatever it takes for Virgil van Dijk
The £48m move broke Liverpool’s transfer record for a second time this summer, following the £37m deal for Mohamed Salah last month.
And The Telegraph add that Jurgen Klopp’s side are prepared to break their transfer record for a third time by spending £70m on Van Dijk.
Liverpool manager Jurgen Klopp recently refused to comment on whether his side will renew their interest in Van Dijk, who cost Southampton just £13m in 2015.
Asked whether they could revive their interest in him, Klopp said: "Am I allowed to talk about this? I am not.
"Pretty much everything I could say would be a mistake - and not because it is an open case or something.
"You or your colleagues wrote everything about this a few weeks ago. I was in Iceland when someone told me it was all in the newspapers.
"Nothing happened since then. All good so far, so I can't say anything about this."
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