We propose a $ SU(3)_C \times SU(2)_L \times SU(2)_N \times U(1)_Y$ model arising from $E_6$ grand unified theory. We show that the tiny neutrino masses in this model can be generated at the three-loop involving the $SU(2)_N$ gauge bosons. With Yukawa couplings around 0.01 or larger and TeV-scale $SU(2)_N$ gauge bosons, we show that the neutrino oscillation data can be explained naturally by presenting a concrete benchmark set of input parameters. All new particles are around the TeV scale. Thus our model can be tested at the ongoing/future collider experiments. This model also has the potential to solve the anomalous magnetic moments of both muon and electron..