The 32-year-old Frenchman has been banished to the sidelines at Stamford Bridge and has not played this season for the club - indeed his last appearance for Chelsea came in the Champions League final.
With the Russian transfer window not due to close until the end of February, Anzhi have time to try and strike a deal for Malouda.
Salavador de Miranda confirmed the interest of Anzhi, he told Izvestia: "At the moment we are working for Florent to leave Chelsea this winter.
"Now we study the offers very carefully and choose the one which has the best conditions for the player. The most important thing is the desire of the player."