Current generation by Thomson scattering in a non-relativistic plasma (kBT/mec2<<1) with the velocity shear and the temperature gradient (Hinata and Daneshvar, 1983) is extended to a relativistic plasma (kBT>mec2) by replacing Thomson cross section by Klein-Nishina formula. Because of the energy dependence of the cross-section, a numerical rather than analytic result is presented. The present calculation may be applied to a supernova implosion where the temperature may reach several MeV and a strong differential rotation is expected. It may also find applications in the early Universe, and laser-pellet interaction.