The 24-year-old has proved himself one of the best defenders in Scotland during his two seasons on Glasgow.
The transfer has been in development for several days, after the Saints beat off a host of Premier League competition to sign the talented central defender.
Sunderland, Everton and Arsenal have all been closely linked to Van Dijk since April according to previous reports in the Mail, but the Dutchman has agreed to meet up with Netherlands legend Ronald Koeman on the south coast.
Van Dijk has made a huge 119 appearances for the Bhoys in just over two seasons since arriving from Groningen, going on to net 15 goals for the Scottish champions.
He won the Scottish Premiership title in both of his campaigns at the club, also picking up the Scottish League Cup in May.
The tall, skilful 24-year-old joins Steven Caulker in trying to plug the hole left from Toby Alderweireld's decision to move to Tottenham this summer, leaving Southampton with some outstanding young defenders to employ alongside Jose Fonte.
Celtic are believed to be about to sign Jozo Simunovic from Dinamo Zagreb to replace Van Dijk.