  We analyze lepton flavor violation (LFV) using the Standard Model Effective
Field Theory (SMEFT) framework at the future lepton colliders. Our focus is on
the associated production of tau lepton with electron/muon at the
electron-positron ($e^+e^-$) colliders, related to four-Fermi SMEFT effective
operators. To respect the upper limits on effective couplings from lepton
flavor violating tau decays, we conduct a cut-based analysis to achieve
sufficient signal significance. We utilize the optimal observable technique
(OOT) to estimate the optimal sensitivity of the effective couplings. The
impact of electron beam polarization and the interplay of signal and background
in enhancing the optimal sensitivity of the effective couplings are discussed
in detail. We find that the sensitivity of flavor-violating effective couplings
is enhanced by order of one for 3 TeV center of mass (CM) energy and 1000
$\rm{fb}^{-1}$ integrated luminosity at the $e^+e^-$ colliders.
