The French midfielder only joined his current club earlier this year, but has long been linked with a big switch to the Premier League after failing to settle with the Russian giants.
Merseyside rivals Liverpool and Everton are believed to be two clubs with a strong interest in the 23-year-old, which has led to rumours of January offers for his signature.
Everton are believed to be the more interested of the two sides, with Roberto Martinez said to be looking at securing a dynamic midfielder with the future of Gareth Barry at the club uncertain.
M’Vila is seen as a perfect addition for the coming years, and the sort of player that can help the Toffees in their pursuit of Champions League qualification.
Liverpool have also been linked with the French international, after pictures of the enforcer among the Reds fans during the most recent Merseyside derby circulated on the internet.
Rubin are not keen to lose one of their star names, but are believed to be willing to listen of offers in the region of £10m-£15m.
Prior to moving to Russia, M’Vila was linked with both Tottenham and Arsenal.