The 24-year-old France international will move to the Italian club on loan for the 2014/15 campaign, with the option to make the switch a permanent one next summer.
M'Vila, who moved to Rubin from Rennes in January 2013, has already undergone a medical at Inter and will officially join the Nerazzurri once the paperwork goes through.
"I only want Inter," he told Sky Sport Italia. "We will not leave Milan until we get the okay from Rubin.
"That is the only thing missing. The medical went very well."
Meanwhile, Inter have announced that they have signed Dodo from Roma on a two-year loan deal.
The 2010 Champions League winners will have the option to sign the Brazilian left-back on a permanent basis in 2016, effectively for a transfer fee of €9m (£7.15m).
Roma have also confirmed the transfer, stating that it will initially cost Inter €1.2million (£950,000) and added: 'This includes an obligatory buy-out clause for €7.8m (£6.20m) to be activated at the player’s first official game."