We present a SQUID susceptometer with a non-homogeneous magnetizing field which is null at the sensing coil and increases towards the patient position with a constant gradient plus a cubic term at large distances. Compared with the magnetizing fields of similar instruments described in the literature, our gradient field enhances the signal due to internal organs with respect to the signal due to superficial tissue. Preliminary measurements have been performed on phantoms of known magnetic susceptibility. The advantage of using a non-homogeneous field compared with a uniform field has been investigated in the case of a double-layer phantom.