The 26-year-old midfielder appears to have been desperate to move to the Stadium of Light after spending last season on loan at the club from Rubin Kazan.
M'Vila made 37 appearances for the Black Cats as he helped them retain their top-flight status, but new manager David Moyes has not opted to sign the Frenchman.
The midfielder spoke of his disappointment in an Instagram post, which he later deleted.
He wrote: "I called Sunderland but they are not picking up the phone. Why? My heart has broken."
M'Vila was left out of Rubin's squad for the league season.