The 26-year-old midfielder made 37 appearances for the Black Cats on a season-long loan last term before returning to parent club Rubin Kazan.
A three-and-a-half year deal had been agreed between Sunderland and M'Vila but the player has now decided against a return to the club.
Black Cats chief executive Martin Bain said: "I was contacted this week by Yann M'Vila's representatives, who advised that the player no longer wishes to join Sunderland.
"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.
"Any player that we bring to Sunderland Football Club must be wholly committed. Yann M'Vila will therefore not be joining us in January and we wish him well for the future."
M'Vila missed just one Premier League game as he helped Sam Allardyce's side avoid relegation last term.