The optical properties of semiconductor quantum wells irradiated by a strong circularly polarized electromagnetic field are studied theoretically. Since the field can induce the composite electron-light states bound at repulsive scatterers, it drastically modifies all optical characteristics of the system. Particularly, it is demonstrated that the quantum interference of the direct interband optical transitions and the transitions through the light-induced intermediate states leads to the Fano resonances in the optical spectra, which can be detected in the state-of-the-art measurements.