Bridcutt has impressed during his time on the south coast since joining from Chelsea in June 2010, having come through the youth academy at Stamford Bridge.
He has played 123 times for Brighton but has not played for the club since New Year's Day after handing in a transfer request.
Poyet appears to have finally got his man though, having tabled an initial offer for 24-year-old Bridcutt and his team-mate Will Buckley earlier in the transfer window.
Sunderland have already signed Marcos Alonso on loan from Fiorentina and Argentinian duo Oscar Ustari and Santiago Vergini in January and are closing in on their compatriot Ignacio Scocco.
Bridcutt's arrival could signal the departure of Lee Cattermole, who has been linked with a move to Stoke, although Potters manager Mark Hughes has insisted no offer has been made.