The 29-year-old spent last season on loan at French side Dijon, following a disappointing 2016-17 campaign as the Black Cats were relegated from the Premier League.
He has not featured in Sky Bet League One this season and a strongly-worded club statement on Wednesday read: "Sunderland AFC has given notice under its contract with Papy Djilobodji.
"Djilobodji, who was under contract to SAFC until June 2020, indicated in June 2018 that he wished to leave the club.
"In order to facilitate that desire, the club entered into a written agreement with the player allowing him to spend the month of July on voluntary unpaid leave.
"When that period came to an end, the player was expected either to leave for a new club - having reached a deal satisfactory to himself and SAFC - or to return in shape to play professional football.
"Instead, he returned to Sunderland over a month later, in the first week of September, ignoring written requests for his return.
"On his return, he was subjected to the same fitness test that his fellow professionals had undertaken on their return. He comprehensively failed that test.
"As a result, Sunderland AFC can confirm that it has accepted Papy Djilobodji's repudiatory breaches of contract and notice of the same has been provided to the player."
Club chairman Stewart Donald was even more critical of Djilobodji and his fellow absentee, record signing Didier Ndong, in an interview on talkSPORT on Tuesday.
"Didier Ndong has shown no interest in returning to the football club whatsoever. We don't even know where he is," said Donald.
"Papy has returned but... there's an obligation on players to turn up in a condition in which you can play football, and the reality of it is that I would probably have beaten his [fitness] stats.
"When you've not turned up for 72 days and missed eight games of the season, and you come up in a condition which means you'll probably miss the next 12 games, I think that sort of says you're not really committed."
