The martensitic transformation temperature and low-temperature aging treatment of Cu-Al-Mn (CAM) shape memory alloys play crucial roles in thermal- and stress-induced martensitic transformations. The local structural changes in Cu induced by low-temperature aging were investigated using X-ray absorption spectroscopy. The average local structure of Cu in the as-quenched CAM alloys is mainly composed of a D03-like structure, which is relatively close to the preparation composition. Furthermore, X-ray absorption near-edge structure spectral analysis of the CAM alloy showed no martensite phase (lattice angle change) was found after the low-temperature aging treatments at less than 150 °C. Instead, we found that the arrangement of Mn atoms changes in two stages during aging treatment.