Despite being heavily linked with a number of Premier League sides, including Arsenal and Tottenham, M'Vila opted to join the Russian outfit in January 2013 from Ligue 1 side Rennes.
However, a change in manager has seen the 23-year-old France international perform well below his best, and current boss Bilyaletdinov admits there have been problems with the midfielder.
"He has to prove his value and work like everyone else," Bilyaletdinov told France Football. "I don't mean that he doesn't work, but we can see that he misses France.
"In January M'Vila behaved badly; one day he trained, one day he didn't. Maybe his agents had promised him something. He thought he was already in Liverpool.
"I told him: 'It is not your level. If you go, you will be on the bench.' He was very disappointed.
"Now I haven't seen him for a while. I have been told he is in France to cure an injury. Even if he is back to full fitness, I don't know where to play him."
M'Vila was pictured at the Merseyside derby with Liverpool fans earlier in the season, fuelling speculation he could join the Anfield side.
He has made 30 appearances in all competitions for Rubin Kazan, registering four assists without scoring.
‘He could have been another Thierry Henry if he’d stayed at Arsenal. It was up to him what he wanted because he was 6ft 1in, quick and unbelievable’: Ray Parlour claims Champions League winner could have enjoyed even greater success with Gunners
‘I don’t think we’ll see Marcus Rashford at Manchester United again – when you’re not seeing eye to eye with a manager, it can hinder your training, but a loan can work, it did for me’ Ex-Premier League star on parallels with forward’s Aston Villa switch
