Listening to speech is difficult in noisy environments, and is even harder when the interfering noise consists of intelligible speech as compared to unintelligible sounds. This suggests that the competing linguistic information interferes with the neural processing of target speech. Interference could either arise from a degradation of the neural representation of the target speech, or from increased representation of distracting speech that enters in competition with the target speech. We tested these alternative hypotheses using magnetoencephalography (MEG) while participants listened to a target clear speech in the presence of distracting noise-vocoded speech. Crucially, the distractors were initially unintelligible but became more intelligible after a short training session. Results showed that the comprehension of the target speech was poorer after training than before training. The neural tracking of target speech in the delta range (1–4 Hz) reduced in strength in the presence of a more intelligible distractor. In contrast, the neural tracking of distracting signals was not significantly modulated by intelligibility. These results suggest that the presence of distracting speech signals degrades the linguistic representation of target speech carried by delta oscillations.