The France international has been on the radars of both clubs for some time after impressing first in his homeland and now in Russia.
Despite being one of Rubin’s top stars, the 23-year-old looks set for the axe at the end of the season, with the club looking to alter their selection and recruitment policies.
It’s understood that big name foreign imports on large salaries are no longer seen as the blueprint for the team, and M’Vila has already been told that he is free to look for a new club.
Liverpool and Everton are likely to be extremely interested in a cut-price deal for the holding midfielder, who in turn is liable to be open to a Premier League switch.
The Reds are perhaps the favourites to land him, with Brendan Rodgers’ side set to book a spot in next season’s Champions League.
M’Vila added fuel to rumours earlier this season after being spotted among the away supporters at Goodison Park during the Merseyside derby.
Everton are also in the hunt, with Roberto Martinez looking to add to his squad as he plots another charge for the top four.