A future high-luminosity $Z$-factory has the potential to investigate lepton flavour violation. Rare decays such as $Z to ell_1^mp ell_2^pm$ can be complementary to low-energy (high-intensity) observables of lepton flavour violation. Here we consider two extensions of the Standard Model which add to its particle content one or more sterile neutrinos. We address the impact of the sterile fermions on lepton flavour violating $Z$ decays, focusing on potential searches at FCC-ee (TLEP), and taking into account experimental and observational constraints. We show that sterile neutrinos can give rise to contributions to BR($Z to ell_1^mp ell_2^pm$) within reach of the FCC-ee. We discuss the complementarity between a high-luminosity $Z$-factory and low-energy charged lepton flavour violation facilities.
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