  Lepton flavor violation is one of the cleanest probes of physics beyond the
standard model. In this work, we explore the sensitivity of the process $e^+
e^- \to \tau \mu$ to new physics above the TeV scale at the proposed circular
electron-positron colliders FCC-ee and CEPC. We compute the $e^+ e^- \to \tau
\mu$ cross-section in the Standard Model Effective Field Theory and assess the
relevant backgrounds. We compare our sensitivity projections to existing and
expected constraints from tau decays and $Z$ decays and find that the future
electron-positron colliders provide competitive probes of new physics. We
highlight the complementarity of searches for resonant $e^+ e^- \to Z \to \tau
\mu$ production on the $Z$ pole and searches for non-resonant $e^+ e^- \to \tau
\mu$ at higher center-of-mass energies.
