We present the results from an experimental study of the magnetotransport of superconducting wires of amorphous indium-oxide having widths in the range 40-120 nm. We find that, below the superconducting transition temperature, the wires exhibit clear, reproducible, oscillations in their resistance as a function of magnetic field. The oscillations are reminiscent of those that underlie the operation of a superconducting quantum interference device.