  A future high-luminosity $Z$-factory will offer the possibility to study rare
$Z$ decays, as those leading to lepton flavour violating final states.
Processes such as $Z \to \ell_1^\mp \ell_2^\pm$ are potentially complementary
to low-energy (high-intensity) observables of lepton flavour violation. In this
work we address the impact of new sterile fermions on lepton flavour violating
$Z$ decays, focusing on potential searches at FCC-ee (TLEP), and taking into
account experimental and observational constraints on the sterile states. We
consider a minimal extension of the Standard Model by one sterile fermion
state, and two well-motivated frameworks of neutrino mass generation, the
Inverse Seesaw embedded into the Standard Model, and the $\nu$MSM. Our study
shows 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 also discuss the
complementarity between a high-luminosity $Z$-factory and low-energy charged
lepton flavour violation facilities.
