By William Joyce
The 25-year-old, who has earned one cap for Argentina, impressed for Gus Poyet's side during a loan spell last season - playing at full-back and in central defence.
Sunderland had hoped to land all three of their loan signings from last season in Fabio Borini, Marcos Alonso and Vergini but, despite bidding for each, only a deal for the latter is currently on the cards.
The Black Cats are now in talks with Estudiantes over a permanent deal and they are confident that he will soon be putting pen to paper on Wearside.
"We know Santi well and he was always one of the first choices," Poyet recently told reporters. "It was him, Marcos and Fabio – all on loan last season – who have been close."
Sunderland are still looking at deals for Alonso and Borini, with Poyet concerned he has only been able to finalise four signings so far this summer and has made his feelings known to the club hierarchy.
Vergini made 11 league appearances last season and was in the side as they pulled off shock wins at Manchester United and Chelsea and drew away to Manchester City to help secure their top-flight survival.