The 32-year-old has not played a single match for the Blues this campaign, and is out of contract at the end of the season. The Frenchman, paid £80,000 to train with the youth team, is keen to leave Chelsea despite the close of the English transfer window.
However the Russian window is still open for a few more days and ex-Chelsea boss Guus Hiddink is set to offer Malouda an escape route, as well as a potential title challenge and a knockout Europa League campaign.
Malouda was one of Jose Mourinho’s last signings at Chelsea when he moved from Lyon for £13.5m in 2007. He has won three FA Cups, the Premier League in 2010 and the Champions League last season, scoring 45 goals in 220 appearances in all competitions for the Blues.
Anzhi are two points behind CSKA Moscow in the Russian league.
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