The 28-year-old is out of contract at the Stadium of Light and head coach Gus Poyet suggested during the season that Bardsley could opt to leave for pastures new.
The defender was pushed out of the team under former boss Paolo Di Canio due to off-field incidents, but he gradually returned to the side.
Bardsley played a role in helping Sunderland to retain their Premier League status by finishing five points above the drop.
According to Sky Sports News, Bardsley is undergoing a medical with fellow top-flight club Stoke, which suggests that the player's six-year stint with the Black Cats is coming to an end.
Bardsley joined Sunderland from Manchester United in 2008.