The 24-year-old midfielder, whose contract is up at the end of the season, was a key player under Gus Poyet during his time as Brighton manager and the Uruguayan views his arrival at the Stadium of Light as a priority.
It would be the second time Poyet has signed Bridcutt, after acquiring the playmaker from Chelsea back in 2010.
Bridcutt is not the only player Sunderland hope to prise from Brighton with the attacker Will Buckley also the subject of an approach from the north-east side.
Poyet revealed on Monday that he hoped to add up to four players to his squad during the winter transfer window, with owner and chairman Ellis Short prepared to back him.
The club have already signed Marcos Alonso from Fiorentina as they prepare themselves for a concerted fight against relegation.