The 26-year-old is currently on loan at the North-East outfit from Costa Rican club Sport Uruguay de Coronado, but the defender has a clause in the deal that can keep him with the Black Cats full-time.
Should Dick Advocaat keep Sunderland in the top flight, a clause will be triggered that allows Vergini to sign a two-year contract.
"I feel very comfortable here - both at the club and outside the club, so I would be very keen to stay," the Sunderland Echo quotes Vergini as saying.
"The only downside is the climate and the weather, but even that has been good for the last few days."
Sunderland are three points above relegation.