Swansea keeper Michel Vorm has agreed personal terms with Tottenham, with Gylfi Sigurdsson set to move the other way.
Championship trio Watford, Wigan and Fulham are all chasing Leicester midfielder Andy King, Sky Sports understands.
M'Vila has been linked with a host of clubs during this transfer window, and Rubin were thought to be in competition with QPR and at least one other Premier League side for his signature.
But Rennes confirmed the 22-year-old is heading for Russia after completing a medical on Wednesday.
"Stade Rennais, FC Rubin Kazan and Yann M'Vila have reached an agreement for the permanent transfer of the French international," said a statement on the Rennes website.
"After the traditional medical visit, the Rennes midfielder joined the Russian club officially on Wednesday evening."
M'Vila has 22 caps for France but is serving an international ban, imposed by the French Football Federation, that will run until June 2014.
M'Vila was punished for taking an unauthorised night out while on duty with the Under-21s last year.
The player has appealed against the length of the ban and could find out this week if he has been successful.