Following speech in noisy and reverberant situations is difficult for cochlear implant (CI) users. This study investigates single- and multi-microphone deep neural network (DNN) speech enhancement algorithms on the joint task of denoising and dereverberation. The DNN algorithms were trained and tested on simulated sound scenes from behind-the-ear hearing devices. Performance was assessed using objective measures and a listening study for reverberant mixtures of speech in multi-talker babble noise. We compare results for signal distortion, predicted intelligibility and speech reception thresholds measured in a listening experiment with 15 typically hearing participants using cochlear implant simulations. Objective metrics indicated listening benefits for both single- and multi-microphone approaches while the listening study results confirmed significant improvements in speech intelligibility for the multi-microphone approaches, holding strong promise to benefit CI listeners.