A listener’s ability to deal with challenging multi-talker situations hinges on their attention resources. While the neural implementation of target enhancement is comparably well understood, processes that enable distractor suppression are less clear. Typically, distractor suppression is quantified by the difference of the behavioural or neural response to distractors versus targets. However, such a difference can be driven by either target enhancement, distractor suppression, or a combination of the two. Here, we designed a continuous speech paradigm to differentiate target enhancement (enhanced tracking of target versus neutral speech) from active distractor suppression (suppressed tracking of distractor versus neutral speech). In an electroencephalography (EEG) study, participants (N = 19) had to detect short repeats in the to-be-attended speech stream and to ignore them in the two other speech streams, while listening also to the content of the to-be-attended audio stream. The ignored speech stream was task-relevant (to-be-attended) in the previous trial and was task-irrelevant in the present trial. The neutral speech stream was always task-irrelevant. We used phase-locking of the EEG signal to speech envelopes to investigate neural tracking via the temporal response function of the brain. Behavioural detection of repeats indicated the suitability of the paradigm to separate processes of attending and ignoring. Sensitivity of behavioral responses according to the Signal Detection Theory revealed that the internal separation for attended versus neutral speech was larger than for attended versus ignored speech. Neurally, the attended stream showed a significantly enhanced tracking response compared to neutral and ignored speech. Unexpectedly, neural tracking did not reveal sizeable differences for neutral versus ignored speech. In sum, the present results show that the cognitive system processes to-be-ignored speech distractors differently from neutral speech. However, this is not accompanied by active distractor suppression in the neural speech tracking response.