Axion-like particles (ALPs) are pseudo-Nambu-Goldston bosons that appear in theories beyond the standard model and are capable of converting to photons in the presence of magnetic fields. Such interaction can have an observable effect on the gamma-ray spectrum of astrophysical targets. We have examined about 40 hours of observations made with the ground-based MAGIC Cherenkov telescopes in the direction of the Perseus Galaxy Cluster. The sources of VHE gamma rays present in the cluster and object of the study are the radiogalaxy NGC1275 and the HBL object IC310. These sources are bright in gamma rays and embedded in the cluster's high magnetic field, making them very good targets for this kind of study. Having not found statistical evidence for ALPs, we exclude ALPs models with masses in the sub-μeV range in line with previous results. Last, we address the potential for gamma-ray-based instruments, both current and future, to identify ALP signals using this analysis technique, finding that the search for wiggles in TeV gamma-ray spectra is very unlikely to be sensitive enough for observations with IACTs.