Liverpool publicly ended their interest in Van Dijk in June after Southampton threatened to report the Reds to the Premier League over an alleged illegal approach for the player.
However, the Merseyside club have continued to be linked with a move for the 26-year-old defender and could launch a formal bid before the transfer window closes.
According to the Evening Standard, Chelsea believe they have a better chance of signing Van Dijk as they have a better relationship with the south coast club than Liverpool.
The Times reported on 3 August that the Blues were preparing a £50m ($66m) plus add-ons bid for the centre-back.
Van Dijk is training on his own at Southampton and was excluded from the club's pre-season tour to France last week.
The Netherlands boss Dick Advocaat gave his support to Van Dijk and urged Southampton to let the defender move to a bigger club.
"You can condemn it, but come on, if you can earn five times as much at a top English club, then I understand you're going to take action," he was quoted as saying by Sky Sports.
Meanwhile, Juventus manager Massimiliano Allegri has said there is "no chance" of Alex Sandro being sold to Chelsea.
"Alex Sandro is a Juventus player and he will stay one," Allegri was reported as saying by ESPN after Juventus' 2-0 defeat to Tottenham at Wembley on 5 August.
Chelsea had a £52m bid rejected for Sandro in June and are reported to be mulling a club-record £61m bid for the defender, along with wages in excess of £100,000 per week.