The Premiership club last night appointed Dutchman Ronald Koeman as their new boss and the former Feyenoord chief has made the Parkhead defender a top target.
Saints stopper Dejan Lovren is attracting interest from a host of top clubs and van Dijk could help fill the gap left by the Croatian international if he is prised away from the South Coast.
The Dutchman caught the eye with a series of superb performances for Celtic last season and could command a fee of around £8m if he was to follow Victor Wanyama in moving from Parkhead to St Mary's.
Hoops chief executive Peter Lawwell has insisted the champions do not have to sell off big name stars, with keeper Fraser Forster also linked with a move away from the club.
But the chance to work with countryman Koeman in the Premiership could sway van Dijk, with the transfer fee he could command then giving Deila room to move in the transfer market.