The Netherlands defender handed in a transfer request at the start of August amid strong interest from Liverpool.
However, a move never materialised before last weekend's deadline as the Saints refused to give in to the player's demands.
Van Dijk is still believed to be pining for a move away and, according to The Mirror, Juve will ask Southampton if a deal can be done to take the Dutchman to Turin at the turn of the year.
The Bianconeri are still searching for Leonardo Bonucci's replacement after their decision to let the Italy international join Serie A rivals AC Milan.
Van Dijk has not played a competitive match since January 22, having missed the closing months of last season with an ankle injury.