Nazi Paikidze, the US women's champion, said: "It is absolutely unacceptable to host one of the most important women's tournaments in a venue where, to this day, women are forced to cover up with a hijab.
Paikidze added: "I am honoured and proud to have qualified to represent the United States in the Women's World Championship. But, if the situation remains unchanged, I will most certainly not participate in this event."
Carla Heredia, an Ecuadorean former Pan American champion, added: "No institution, no government, nor a Women's World Chess Championship should force women to wear or to take out a hijab.
"This violates all what sports means. Sport should be free of discrimination by sex, religion and sexual orientation."
Susan Polgar, the chairman of Fide's commission on women's chess, defended the federation, saying that women should respect "cultural differences".