We discuss a possibility that the neutron electric dipole moment (NEDM) can be calculated in lattice QCD simulations in the presence of the CP-violating theta term. In this paper we measure the energy difference between spin-up and spin-down states of the neutron in the presence of a uniform and static external electric field. We first test this method in quenched QCD with the renormalization group improved gauge action on a 16(3)x32 lattice at a(-1)similar or equal to 2 GeV, employing two different lattice fermion formulations, the domain-wall fermion and the clover fermion for quarks, at relatively heavy quark mass (m(PS)/m(V)similar or equal to 0.85). We obtain nonzero values of the NEDM from calculations with both fermion formulations. We next consider some systematic uncertainties of our method for the NEDM, using 24(3)x32 lattice at the same lattice spacing only with the clover fermion. We finally investigate the quark mass dependence of the NEDM and observe a nonvanishing behavior of the NEDM toward the chiral limit. We interpret this behavior as a manifestation of the pathology in the quenched approximation.