Sunderland said they agreed a three-and-a-half-year deal with M'Vila but "decided not to enforce the contract".
M'Vila, 26, made 37 appearances for the Black Cats on a season-long loan last term before returning to Rubin Kazan.
"Any player we bring to Sunderland must be wholly committed," said chief executive Martin Bain.
M'Vila missed just one Premier League game as he helped Sam Allardyce's side avoid relegation last term, with Black Cats fans hopeful the club would sign him on a permanent deal last summer.
However, the move failed to materialise on transfer deadline day in September, leading to M'Vila posting an Instagram message - which was later deleted - saying he had tried contacting the club to push the deal through.
M'Vila returned to the Russian club when his loan deal expired, making nine appearances in the Russian Premier League this season.
Sunderland had not announced they had agreed a deal to re-sign the player until they revealed on Wednesday it had fallen through.
"I was contacted this week by Yann M'Vila's representatives, who advised that the player no longer wishes to join Sunderland," added Bain.
"There was a three-and-a-half-year agreement in place, however after discussions with the manager we have decided not to enforce the contract."
Sunderland say they will not be signing Yann M'Vila in January after being told the French midfielder no longer wants to re-join the Premier League club.