The Reds are expected to delve into the transfer market in the coming weeks to bolster their squad, with Brendan Rodgers keen to maintain a push for Champions League qualification.
One area of his side the Northern Irishman is looking to strengthen is the centre of midfield, with a physical presence needed.
M’Vila is seen as an ideal solution, and a player that can be secured before the deadline passes.
Despite having only moved to Russia 12 months ago, the 23-year-old is believed to be desperate to move back to one of Europe’s top leagues in a bid to further his hopes of returning to the French national side on a consistent basis.
M’Vila was linked with Arsenal and Tottenham prior to his big money transfer to Rubin, at which time he was being hailed as one of the continent’s top emerging talents.
The Merseysiders may push for a loan deal with a view to a permanent switch at the end of the season, with Rodgers keen to assess the enforcer in more detail over the course of the season.