The Rubin Kazan man spent last season on loan with the Black Cats and played an important role in helping them stave off relegation under Sam Allardyce’s guidance.
The midfielder, 26, made 40 appearances in all competitions for the Wearsiders, scoring once, and he could be set to return to the Stadium of Light.
The Sunderland boss – who has been linked with the England job in recent days – has reportedly travelled back from the club’s pre-season training camp to ensure the deal for M’Vila gets over the line.
However, according to Foot Mercato, the Frenchman is being chased by Koeman’s Toffees, who are ready to pay over the odds to land him.
While Kazan are keen to keep hold of highly-rated midfielder M’Vila, they will reluctantly sell if the right offer comes in, rather than risk losing him on a free when his contract expires next summer.