The Sweden international has been considering his options since Birmingham's relegation from the Premier League.
The 26-year-old midfielder turned down a move to the continent, despite interest from La Liga, to remain in England.
Fulham, Aston Villa and Arsenal are all believed to have made offers, but he is believed to have opted to make the switch to the North-East.
Sunderland boss Steve Bruce signed him for Birmingham and has agreed a deal to take him to the Stadium of Light.
His former club Arsenal had offered him the chance to return to North London but knowing he would be a squad player with the Gunners was seemingly enough to put him off the move.