The 23-year-old has made just eight first-team appearances for the Blues since arriving from PSV Eindhoven in 2007 and is entering the last 12 months of his contract.
Van Aanholt has been sent on loan to no fewer than five clubs and is determined to quit Stamford Bridge this summer, stating: "Where will I play next season? I don't know yet. It definitely won't be at Chelsea. I am done here."
Sunderland appear set to win the chase for the Dutch international having so far failed to prise last season's loan star Marcos Alonso away from Fiorentina on a permanent basis.
Manager Gus Poyet has already completed the Bosman signings of Billy Jones, Costel Pantilimon and Jordi Gomez for next term but remains keen to continue increasing his defensive options.