In this work we numerically calculate cross sections of the reactions 6Li(n,n′γ)6Li and 7Li(n,n′γ)7Li reactions in the region of neutron kinetic energy En=0.5−20 MeV. Neutron-nucleus interaction is defined in terms of the three-body problem wherein the target-nucleus is a bound two-body system. Dynamics in all two-body segments is described by the coupled Lippman-Schwinger equations with separable potentials. Free parameters of these potentials are chosen based on a fairly good description of elastic scattering phase shifts and radiation cross sections: tα→γ7Li, dα→γ6Li, nd→γt. The system of three-body strongly interacting particles is described by the Faddeev equations. Radiative relaxation of the excited nucleus is effectively taken into account by the electromagnetic vertex inserted in the three-body scattering amplitude. Total cross sections and also poorly measured double differential cross sections of 6Li(n,n′γ)6Li and 7Li(n,n′γ)7Li are obtained and briefly analyzed.