We consider the scenario in the radiative seesaw model where the dark matter particle is the lightest $Z_2$-odd fermion. We identify the regions of the parameter space of the model compatible with neutrino oscillation data, with the upper limits from rare charged lepton decays and with the observed dark matter abundance via thermal freeze-out, and we compute the dark matter scattering cross section with nuclei via the one-loop exchange of a photon, a $Z^0$-boson or a Higgs boson. We find that the predicted spin-independent cross section lies below the current LUX limit, although, for some choices of parameters, above the expected sensitivity of XENON1T or LZ.