Many extensions of the Standard Model of particle physics such as superstring and superbrane theories predict the existence of axion-like particles (ALPs). ALPs are very elusive particles, which primarily interact with photons and in the presence of an external magnetic field two effects are produced: (i) photon-ALP oscillations, (ii) change of the polarization state of photons. The astrophysical context represents the best possibility to obtain indirect evidence for the ALP existence thanks to the various effects that photon-ALP interaction produces. Great attention has been paid so far to the former effect: photon-ALP interaction modifies the transparency of the crossed media and final spectra are altered presenting either a flux excess or irregularities or both these features. Two hints at ALP existence have also been discovered. However, less interest has been attracted by the latter main ALP effect: the modification of photon polarization. In this paper we want to address it both in the X-ray and in the high-energy (HE) bands. In particular, we analyze the photon degree of linear polarization and the polarization angle when photon-ALP interaction is present concerning photons generated in the central region of galaxy clusters: we study Perseus and Coma. We observe that photon-ALP interaction substantially changes the photon polarization expected from standard physics and that it shows interesting features both in the X-ray and in the HE band. We conclude that the ALP-induced polarization effects are more likely detectable from the proposed missions like COSI, e-ASTROGAM and AMEGO in the HE range.