Chelsea have kept their long-standing interest in Van Dijk, while Arsenal are now ready to enter the fray, and both clubs are considering late bids - though the problem would then be that Southampton accepting any offer would then allow Liverpool back in to match it.
It is said that sources close to the player insist Van Dijk, 26, expects to be a Liverpool player in little over 48 hours "and that it is still by far his first preference".
Arsene Wenger and Antonio Conte are expected to be extremely active during the final days of the window and van Dijk would obviously be a sensational signing if either club could get a deal agreed.
Arsenal have been linked with a move for Van Dijk - as part of a swap-deal involving Theo Walcott - to bolster their shaky back-line.
That's after the Reds were forced to issue a public apology earlier in the summer for illegally tapping up the Dutch centre-half.
Southampton have made it clear publicly and privately that they will not consider selling Van Dijk, who has been training alone since he handed in a transfer request three weeks ago.
The Netherlands global, who has not made a Premier League appearance so far this season, still has five years remaining on his current contract with Southampton.
But his situation has not caused rancour within the dressing room, according to Southampton captain Steven Davis.
'We will welcome him back. Hopefully we'll keep him as it will make our squad stronger'.