The Blues, along with Premier League rivals Liverpool, are thought to be the front-runners in the race to sign the 12-time capped worldwide, though Southampton favour a sale to the former due to Liverpool's rumoured illegal approach for Van Dijk earlier this summer.
The Netherlands global defender has been quoted as saying by BBC Sport that he would like Southampton to consider interest in him from top clubs.
'It is with regret I can confirm I have handed in a transfer request, ' read a statement from Van Dijk, also linked with Chelsea.
"I am incredibly ambitious and want to achieve as much as I possibly can to fulfill my potential in what is a very short career as a professional footballer", he added.
Liverpool avoided a sanction, but were forced to abandon their move for the defender and make a public apology for their actions.
The Anfield club said they were no longer interested in him again and Klopp has reiterated that.
The former England worldwide also explained that while Chelsea would look the more appealing opportunity given they are reigning Premier League champions, the presence of David Luiz may mean that Van Dijk is played out of position, rather than the centre stage he would be offered at Anfield. Five. Then you win a prize.
"In the moment I'm fine".
Liverpool manager Jurgen Klopp is satisfied with his current defensive options and believes there are few centre backs in the transfer market who could improve his current squad ahead of the Premier League season. We can not force the things and we would never get a centre back just [for the sake of it].
As an ex-pro I would have thought you'd see that. I think only one player is not enough to improve our team, ' said Conte.