Lepton flavor violating decays of the Higgs boson $h$ could appear in scenarios beyond the Standard Model of particle physics. In this article, we study the sensitivity of a future Compact Linear Collider (CLIC), operating in $e^{+}{e}^{-}$ collisions, to such processes, namely, $h\to e\mu$, $h\to \tau \mu$, and $h\to e\tau$. In the absence of any observation, 95% CL upper limits in the region of ${10}^{-4}$ to ${10}^{-5}$ could be placed on the branching fractions of these processes, assuming integrated luminosities of $4$,ab${}^{-1}$ at $\sqrt{s}=1.4$ TeV and $5$,ab${}^{-1}$ at $\sqrt{s}=3$ TeV.