Musical rhythm positively impacts on subsequent speech processing. However, the neural mechanisms underlying this phenomenon are so far unclear. We investigated whether carryover effects from a preceding musical cue to a speech stimulus result from a continuation of neural phase entrainment to periodicities that are present in both music and speech. Participants listened and memorized French metrical sentences that contained (quasi-)periodic recurrences of accents and syllables. Speech stimuli were preceded by a rhythmically regular or irregular musical cue. Our results show that the presence of a regular cue modulates neural response as estimated by EEG power spectral density, intertrial coherence, and source analyses at critical frequencies during speech processing compared with the irregular condition. Importantly, intertrial coherences for regular cues were indicative of the participants' success in memorizing the subsequent speech stimuli. These findings underscore the highly adaptive nature of neural phase entrainment across fundamentally different auditory stimuli. They also support current models of neural phase entrainment as a tool of predictive timing and attentional selection across cognitive domains.