Wright played six matches for the League One side last season, helping them keep five clean sheets, while on loan from Championship side Bristol City.
Unfortunately for Wright, his playing time was curtailed by an ankle injury and the premature end to the season caused by the coronavirus pandemic.
The 28-year-old has played 24 times for Australia, with his last appearance coming against Nepal in October 2019.
“It feels really good to get the deal done and I just can’t wait to get up to the North East and join the squad," Wright said.
"Although I only worked with the manager for a short spell, I really enjoyed the structure and discipline of the team.
"He expects his players to give everything on the pitch and be good people, and we have that culture within the dressing room.
"We are heading in the right direction and I’m very confident we can achieve our ambitions next season.”
Sunderland finished eighth last season in League One.