The Netherlands international was a target for the Reds during the summer, only for a rumoured £50m deal to break down because of an illegal approach.
Liverpool were forced to publicly apologise to the Saints and did not return with a fresh offer in the closing stages of the window, leaving Van Dijk's Southampton career in limbo.
Manchester City are also understood to have been circling, without following through with their interest, but The Mirror reports that Barcelona have serious aspirations of signing the former Celtic defender.
Van Dijk is said to have been watched by Barca scouts during his country's 2-0 win against Sweden earlier this week, impressing enough for a January bid to be lined up.
The report suggests that Southampton would prefer to offload to a European side rather than a domestic rival.