Axions and axion-like particles emerge in many models for physics beyond the Standard Model. Thus, they have gained increasing research interest in both experimental and theoretical physics apart from their original proposition as a solution to the strong CP-problem. Among other aspects it has recently been shown that ALPs can potentially provide a solution to the long-lasting discrepancy between theory and experiment of the anomalous magnetic moment of the muon. Provided that the ALP has flavor-violating couplings to leptons, they can also mediate flavor-violating decays like $$\mu \rightarrow e\gamma $$ $\mu \to e\gamma$ . Both processes are mediated through related form factors that we compute to two-loop order. We further show numerical implications of our calculations and how they might affect constraints on ALP couplings derived from experiments.