The 25-year-old is one of the hottest properties on the transfer market.
Liverpool boss Jurgen Klopp has made his signing a top priority.
The Reds refused to comment on renewed speculation but it is expected they will intensify efforts to reach a deal with both Southampton and the player.
Chelsea and Manchester City have both been heavily linked with the defender, who joined Southampton from Celtic for £13m in September 2015.
But BBC Sport understands Van Dijk wants to move to Anfield in what would be a major coup for Klopp and Liverpool owners Fenway Sports Group, who are willing to spend big after qualification for next season's Champions League.