Gennadii Onishchenko, the Health Ministry's chief health inspector, told Russia's Ekho Moskvy radio station that authorities fear population movements connected with the annual Muslim pilgrimage to Mecca may help the deadly virus spread further.
Onishchenko is the chairman of Rospotrebnadzor, the government agency responsible for monitoring the quality of food products reaching Russian consumers.
Authorities in Turkey say 14 people have been infected with the bird-flu virus, including three fatalities.
In Georgia and Azerbaijan, governments are taking preventive steps to prevent the virus from entering their respective countries. They say no bird-flu cases have been recorded so far.
Turkey shares common land borders with Georgia and Azerbaijan's autonomous exclave of Nakhichevan.