The Reds were after the Dutchman earlier in the summer transfer window, however, they had to end their interest in Van Dijk in June following a formal apology to the Saints for allegedly "tapping up" the player.
Despite the setback, Liverpool have been repeatedly urged to pursue the 26-year-old rated at £60m ($76.8m) who earlier in August, submitted a transfer request.
While Saints chairman Ralph Krueger has recently come out and claimed that the defender will not leave St. Mary's this year at least, Aldridge believes there is still a possibility of him leaving with less than a week to go in the transfer window.
"In an ideal world, the deal to get Virgil van Dijk out of Southampton could be revived and their signing of a Dutch international defender Wesley Hoedt suggests they may be covering themselves in case their 'on strike' centre-back leaves," Aldridge wrote in his Independent column.
"Liverpool appear to have made a bit of a mess of their initial approaches for Van Dijk, with their apology to Southampton for speaking to the player without permission appearing to have soured relations between the two clubs.
"Yet there comes a point when a player who is causing too many problems at a club needs to be removed and with Van Dijk making it clear he wants to leave, there may be room to strike a deal if a buying club pays over the odds."
Aldridge also believes comparisons with Philippe Coutinho are unwarranted as despite what is increasingly looking like a failed attempt at moving to Barcelona, the Brazilian is likely to accepted by everyone at the club if he stays, unlike Van Dijk.
"If Coutinho comes back into the Liverpool fold once the transfer window closes, I don't think the players, the manager or the fans will have a problem with him, but you wonder if that is the same story with Van Dijk at Southampton," he added.
"Van Dijk is at loggerheads with his employer and that is not a healthy situation. I can see something breaking there in the next few days and a transfer happening.
"Whether Southampton would sell Van Dijk to Liverpool after all that has gone on this summer is another matter, but they should certainly make an offer in the next few days just to test the water."