Team director and Brazilian legend Roberto Carlos said Anzhi want to link Rooney, 27, with Cameroon striker Samuel Eto'o, who is currently their top earner at 350,000 pounds a week.
Billionaire Russian oligarch Suleyman Kerimov owns the Dagestan based club.
Asked if Anzhi might sign Rooney, Carlos said: "Why not? He's one of the best players in the world and any team would be happy to have him in their squad. He's still a very young player.
"Of course, I could not say when and if it will happen. It's not only Anzhi who want to buy him. There are plenty of teams in Italy and Spain, all over the world, eager to have a player like Rooney."
Rooney would also be linking up with ex-Chelsea chief Guus Hiddink, who joined Anzhi this season and has guided them to the top of the Russian Premier Division, The Sun reports.
Carlos said: "If the owner has a wish to buy somebody, he will go and do it. But we are trying to change the image of the club. It is not just about money."
The Dagestan side trains near Moscow and make a 2,000-mile round trip for home games. (ANI )