Axion-like particles (ALPs) and photons inter-convert in the presence of a magnetic field. At keV energies in the environment of galaxy clusters, the conversion probability can become unsuppressed for light ALPs. Conversion of thermal X-ray photons into ALPs can introduce a step-like feature into the cluster thermal bremsstrahlung spectrum, and we argue that existing X-ray data on galaxy clusters should be sufficient to extend bounds on ALPs in the low-mass region $m_a lesssim 1 times 10^{-12},{rm eV}$ down to $M sim 7times 10^{11}, {rm GeV}$, and that for $10^{11}, {rm GeV} < M lesssim 10^{12}$ GeV light ALPs give rise to interesting and unique observational signatures that may be probed by existing and upcoming X-ray (and potentially X-ray polarisation) observations of galaxy clusters.