Arsenal and Chelsea are also interested but the Southampton centre-back wants to join the Reds before the window slams shut, according to the Independent.
Saints chiefs are unwilling to sell to Liverpool as a result of the bad blood between the clubs.
Liverpool were dramatically forced to ditch a move for Van Dijk earlier this summer after Southampton caught wind of an alleged illegal approach.
They threatened to report them to the Premier League so Anfield bosses publicly apologised.
Liverpool are reportedly set to launch a club-record £75m bid for Van Dijk ahead of Thursday’s deadline.
But they are likely to face competition for his signature.
According to the Independent, Chelsea and Arsenal are considering late bids.
Liverpool boss Jurgen Klopp is desperate to bring Van Dijk to Merseyside to bolster his flimsy defence.
Van Dijk has been frozen out by Southampton head coach Mauricio Pellegrino since handing in a transfer request last month.
