The Dutch centre-back has informed Southampton that he wishes to leave the south coast club and has been training on his own, although Saints officials have insisted that he will not be sold this summer.
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.
Chelsea's optimism in signing Van Dijk stems from the fact that the Saints have a poor relationship with Liverpool and they might be more open to letting him move to Stamford Bridge.
Southampton boss Mauricio Pellegrino reiterated on 3 August that the club would not allow Van Dijk to join Liverpool.
"We continue in the same way. The situation is the boy is aside of the team because he doesn't want to play for us," Pellegrino was quoted as saying by the Daily Telegraph.
"I would like to persuade him that he's important for us and the club will not sell him."
The Saints signed Van Dijk from Scottish club Celtic for £13m in 2015. He is contracted to the south coast club until 2022.
Chelsea have already brought in defensive reinforcements ahead of the new season in the form of Antonio Rudiger from Roma, while Tiemoue Bakayoko, Willy Caballero and Alvaro Morata have been other additions.
The Blues kick-start their season with a Community Shield clash against Arsenal at Wembley on 6 August.