That’s according to respected Merseyside football journalist Paul Joyce, who says Saints knew the Reds were willing to pay £70m for the Holland international.
Van Dijk emerged as the man Klopp wanted to bolster his defence towards the end of the 2016/17 campaign.
But Liverpool were forced to officially end their interest in signing the 26-year-old after being accused of tapping him up.
Liverpool were expected to make a deadline day swoop for Van Dijk anyway - although that failed to materialise.
Joyce says Klopp remains keen on getting the former Celtic man on-board though.
And, writing for The Times, he’s suggested Liverpool will pursue a deal in future.
Van Dijk made it clear he wanted to quit Southampton by handing in a transfer request.
And Anfield was known to be his preferred destination, amid Arsenal, Chelsea and Manchester City also monitoring him.
Southampton were unrelenting in their stance that he would not be sold this summer.
However, that could alter if Van Dijk refuses to reintegrate into first-team affairs.
Southampton were aware that Liverpool were willing to pay a world-record defender fee of £70m for Van Dijk.
And that figure may tempt St. Mary’s chiefs in future if they cannot persuade him he should stick around for the long-term.
