Lepton flavor violating Higgs boson decays offer a direct probe of physics beyond the Standard Model. We summarize a full-simulation study of the sensitivity of the Compact Linear Collider (CLIC) to $h\to e\mu$, $h\to\tau\mu$ and $h\to e\tau$. The analysis considers Higgs production through $WW$ fusion at center-of-mass energies of 1.4 and 3 TeV, with integrated luminosities of 4 and 5 ab$^{-1}$, respectively, and unpolarized beams. Multivariate analyses yield projected 95\% confidence level upper limits on the branching fractions between $1.44\times10^{-5}$ and $1.69\times10^{-4}$, highlighting the potential of high-energy lepton collisions to investigate charged lepton flavor violation.