Former Manchester United and Everton boss Moyes is poised to replace Sam Allardyce in the dugout at the Stadium of Light once he is officially appointed as the new England boss.
Moyes is a big fan of M'Vila, having previously tried to sign the midfielder in January 2013 back when he was Everton boss.
Everton and QPR were battling for the then-Rennes midfielder's signature before he eventually signed for Rubin Kazan.
M'Vila was a key figure for Sunderland last season, making 40 appearances for the Black Cats, and has come out and said he wants to return to the club this summer.
Kazan see no future for the French international at the club and he was not included in the squad that flew to Austria earlier this month for a pre-season training camp.
And M’Vila, along with compatriot Chris Mavinga - also surplus to requirements after spending last season on loan – has now been given permission to return to his homeland.
Kazan claim this is due to a visa issue after the pair returned late to pre-season training last month due to paperwork issues.
But there have been reports in France that M’Vila and Mavinga were not happy at being left to train alone, or with the training facilities at the club.