The club has written an open letter to UEFA president Michel Platini saying it was "deeply sorry and surprised" to hear that it would not be able to stage Europa League matches at its base in the violence-plagued Dagestan region.
UEFA, which according to Anzhi's letter made the decision for security reasons, was not immediately available for comment.
"Anzhi has been hosting its home matches in the Russian championship in Makhachkala for more than 20 years. During this time, no incident of any gravity has happened in our stadium," said the letter written by general director Aivaz Kaziakhmedov.
"The stars of the European and world football, Roberto Carlos, Samuel Etoo, Yuri Zhirkov, and others enjoy playing for Anzhi in Makhachkala under the leadership of Guus Hiddink.
"Millions of our fans were waiting for a European match in Makhachkala as a real festival.
"The owner of the Anzhi club has invested millions to finance the reconstruction of the stadium, building of a new ultra-modern football arena in the city, youth football academy and the construction of seven football centres."
"After such a decision made by UEFA, what shall we do with all these projects?
"There is just one question left - maybe we'd better stop this initiative, maybe all this is useless for the football development and the formation of healthy life-style, solution of the citizens' social problems."
Anzhi will enter the tournament in the second qualifying round and will meet the winners of the tie between Flamutari of Albania and Hungarians Honved.