The Celtic star recently spoke of his desire to move on to a new challenge, as well as his disappointment that the Bhoys have refused to sell him in the past.
And Advocaat is reportedly keen enough on signing the 23-year-old that he is willing to let one of his current defenders leave to make room for his fellow countryman.
Reports from the Daily Mail earlier this year had claimed Vergini will be offered a two-year contract extension if Sunderland stay up.
However, the Sunderland boss is said to prefer Celtic's Van Dijk - although he could face competition from fierce rivals Newcastle United, the Telegraph add.
Wes Brown has signed a new deal at the Stadium of Light but Advocaat is still looking to strengthen his back line, which is no surprise given the Black Cats' woeful defensive record in recent seasons.