Dealing with speech interference in a speech enhancement system requires either speaker separation or target speaker extraction. Speaker separation has multiple output streams with arbitrary assignments while target speaker extraction requires additional cueing for speaker selection. Both of these are not suitable for a standalone speech enhancement system with one output stream. In this study, we propose a novel training framework, called Attentive Training , to extend speech enhancement to deal with speech interruptions. Attentive training is based on the observation that, in the real world, multiple talkers very unlikely start speaking at the same time, and therefore, a deep neural network can be trained to create a representation of the first speaker and utilize it to attend to or track that speaker in a multitalker noisy mixture. We present experimental results and comparisons to demonstrate the effectiveness of attentive training for speech enhancement.