It is believed the France midfielder is keen to move to the Premier League club and could undergo a medical within the next 24 hours.
Sky Sports reported earlier this month that Sunderland were interested in M’Vila as well as QPR midfielder Leroy Fer, who has since returned to Loftus Road after failing a medical.
M’Vila signed for Rubin for £10m from French Ligue 1 side Rennes two-and-a-half years ago, having reportedly been the subject of interest from a number of top European clubs.
He spent last season on loan at Inter Milan and looked set for another temporary move to Dinamo Moscow after a reported dispute with his parent club.
But Sunderland are set to land M’Vila and add him to their list of summer signings Jeremain Lens, Younes Kaboul, Ricky Alvarez, Adam Matthews and Sebastian Coates.
M’Vila has 22 caps for France, having made his international debut in 2010, but in November 2012, he was banned from national team matches until June 2014 after an unauthorised night out while on U21 duty.
He has not made an international appearance since.