This paper introduces a novel neural network-based speech coding system that can process noisy speech effectively. The proposed source-aware neural audio coding (SANAC) system harmonizes a deep autoencoder-based source separation model and a neural coding system so that it can explicitly perform source separation and coding in the latent space. An added benefit of this system is that the codec can allocate a different amount of bits to the underlying sources so that the more important source sounds better in the decoded signal. We target a new use case where the user on the receiver side cares about the quality of the non-speech components in speech communication, while the speech source still carries the most crucial information. Both objective and subjective evaluation tests show that SANAC can recover the original noisy speech better than the baseline neural audio coding system, which is with no source-aware coding mechanism, and two conventional codecs.