  Any observation of charged lepton flavor violation (CLFV) implies the
existence of new physics beyond the SM in charged lepton sector. The muon
magnetic moment anomaly between the SM prediction and the recent muon $g-2$
precision measurement at Fermilab provides an opportunity to reveal the nature
of underlying CLFV. We consider the most general SM gauge invariant Lagrangian
of $\Delta L=0$ bileptons with CLFV couplings to bilinear leptonic fields. The
muon $g-2$ anomaly constrains the CLFV coupling $y^{\ell\ell'} (\ell\neq
\ell')$ which induces the leptonic transition with
$|\Delta(L_\ell-L_{\ell'})|=4$ at tree level. We investigate the implication of
the muon magnetic moment anomaly for the CLFV couplings and other relevant
constraints from low-energy precision experiments. The projected sensitivity of
the future muonium-antimuonium conversion experiment and colliders is evaluated
to the individual CLFV couplings satisfying these constraints.
