Properties of a strongly-coupled nonlinear directional coupler (NLDC) with a lossy MQW coupling layer is analyzed using the Galerkin finite element method accompanied by a predictor-corrector algorithm. It is shown that the propagation attenuation along the NLDC is considerably smaller than that in the bulk MQW and tends to reduce with the input power. By the presence of losses, the powers guided in two waveguides do not become a maximum and a minimum at the same propagation length, unlike the lossless coupler. The losses make the nonlinear effect weak due to the decrease in guided power, and hence the coupling length decreases and the switching power increases. The extinction ratio of the switching becomes the largest value not in the cases of nonloss and high losses but in the case of moderately high losses, although the switching power is somewhat larger than that of the lossless case.