Seeing a person talking can help to understand them, in particular in a noisy environment. However, how the brain integrates the visual information with the auditory signal to enhance speech comprehension remains poorly understood. Here we address this question in a computational model of a cortical microcircuit for speech processing. The model consists of an excitatory and an inhibitory neural population that together create oscillations in the theta frequency range. When simulated with speech, the theta rhythm becomes entrained to the onsets of syllables, such that the onsets can be inferred from the network activity. We investigate how well the obtained syllable parsing performs when different types of visual stimuli are added. In particular, we consider currents related to the rate of syllables as well as currents related to the mouth-opening area of the talking faces. We find that currents that target the excitatory neuronal population can influence speech comprehension, both boosting it or impeding it, depending on the temporal delay and on whether the currents are excitatory or inhibitory. In contrast, currents that act on the inhibitory neurons do not impact speech comprehension significantly. Our results suggest neural mechanisms for the integration of visual information with the acoustic information in speech and make experimentally-testable predictions.