A new method for searching for Dark Matter axions is proposed. It is shown that a two-contact SQUID can detect oscillating magnetic perturbations induced by the axions in a strong inhomogeneous magnetic field. A resonant signal is a steplike response in the SQUID current-voltage characteristic at a voltage corresponding to the axion mass with a height depending on the axion energy density near the Earth. The proposed experimental technique appears to be sensitive to the axions with masses ma≲10−4 eV, which is well-motivated by current researches both in cosmology and in particle physics.