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.
Liverpool still have to reach an agreement with Southampton - and a demand for a fee well in excess of £50m could mean negotiations will be complicated before any deal nears a conclusion.
Former club Celtic are also understood to have a 10% sell-on agreement.
Van Dijk missed the final five months of the season after sustaining an ankle injury following a challenge by Leicester City's Jamie Vardy in Southampton's 3-0 win at St Mary's on 22 January.
It forced him to miss Southampton's EFL Cup final loss to Manchester United at Wembley.
Liverpool are poised to win the race to sign Southampton's £50m-rated Netherlands defender Virgil van Dijk after it emerged he wants to move to Anfield if he leaves St Mary's.