  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.
