With the ever-growing demand for deep learning (DL) at the edge, building small and efficient DL architectures has become a significant challenge. Optimization techniques such as quantization, pruning or hardware-aware neural architecture search (HW-NAS) have been proposed. In this paper, we present an efficient HW-NAS; Compression-Aware Neural Architecture search (CaW-NAS), that combines the search for the architecture and its quantization policy. While former works search over a fully quantized search space, we define our search space with quantized and non-quantized architectures. Our search strategy finds the best trade-off between accuracy and latency according to the target hardware. Experimental results on a mobile platform show that, our method allows to obtain more efficient networks in terms of accuracy, execution time and energy consumption when compared to the state of the art.