A model explaining the presence of a periodic train of microwave absorption lines in the magnetic modulated microwave absorption (MMMA) spectra of high- and low-temperature superconductors is proposed. The model assumes the occurrence of regular superconducting current loops, closed by Josephson junctions, in these materials. The system of such loops is considered within the basic model of the rf superconducting quantum interference device taking into account the effect of thermal fluctuations. The magnetic-field and temperature dependencies of the MMMA obtained on the basis of the proposed model are in qualitative agreement with experimental data.