The Black Cats have agreed a deal with the Argentinian full-back's current club, Estudiantes, and have agreed personal terms with Angeleri and his representatives.
However, the transfer will not be confirmed until Angeleri has received an Italian passport, which will remove the need for a work permit to play in the Premier League.
Angeleri is one of two right-backs due to arrive on Wearside in the next few days.
Ahmed al-Muhammadi should undergo a medical in the North-East later today after Sunderland officials agreed a season-long loan with Egyptian side ENPPI, with a view to a permanent transfer next summer.
The moves cast a major doubt over Phil Bardsley's future at the Stadium of Light.