The 25-year-old, who won the Argentins title with Newell's Old Boys last season, is set to become the Black Cats' second January signing, with Mariano Andujar also a target
Vergini, who won the Argentine title with Newell's Old Boys before joining their rivals in the close season, will become the Black Cats' second signing of the month after Fiorentina full-back Marcos Alonso joined on loan for the remainder of the season.
"He wants to leave Estudiantes, and as he isn't a player that is considered as principle in the squad, we have accepted his departure," club president Enrique Lombardi told Diario Ole of Vergini.
"He wants to change team, and as there is a good offer that we have analysed, he will leave our club."
And when asked about the possibility of a deal, Poyet replied: "With Vergini, we're close. I've talked to him and it's well on the way. I've known him for years."
The Black Cats boss also revealed he is interested in acquiring the services of Catania goalkeeper and Argentina international Mariano Andujar.
He added: "I like the style of South American goalkeepers and Andújar is a player that I like and I know."