Aiden McGeady, Luke O’Nien and Denver Hume have also been offered new contracts.
Wyke is one of four players, alongside Aiden McGeady Luke O’Nien and Denver Hume, who have been offered new deals by the club.
Meanwhile Max Power and Grant Leadbitter, both of whom captained the Black Cats at some stage last season, are among seven senior players to be released.
Sunderland’s bid to earn promotion back to the Championship came to an end in their play-off semi-final defeat to Lincoln on Saturday, though the club did win the Papa John’s Trophy in March.
Boss Lee Johnson told the club’s official website: “The players have worked extremely hard this season in very difficult circumstances and their reward for that was a trophy-winning season and a play-off spot.
“We are all disappointed we couldn’t achieve our ultimate goal of winning League One, but we thank the players for their endeavours and commitment and we will always welcome them back to the football club with open arms.
“As we look to a new dawn, there are many reasons to be optimistic about our future as we continue to rebuild the football club.”
The additional players to be released alongside Power and Leadbitter are Callum McFadzean, Chris Maguire, Conor McLaughlin, Josh Scowen and Remi Matthews.