For the 9/2-[514] bands in light odd-A Re isotopes, the energy signature splitting and its relation with the signature dependence of M1 transition matrix elements are investigated in connection with the deviation of nuclear shape from axial symmetry. By comparing the energy signature splittings and relative magnetic transition rates between the experimental values and the theoretical calculations assuming axially symmetric shapes, it is found that discrepancies increase with the decreasing neutron number. These discrepancies strongly suggest an appreciable negative γ deformation for the very neutron-deficient odd-A Re isotopes.