The Argentine defender arrived on Wearside last year and he went on to make over 50 appearances for the club.
However, Advocaat has conceded that the 26-year-old is not a part of his plans for the forthcoming season.
"It's a new season and I have to be straight and honest with people. I don't want to go on with him, with all respect. I'm sorry for Santi, but he has to go and find another club," the Dutchman told reporters.
Vergini is currently contracted to Estudiantes in his homeland.