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We have examined the anomalous electromagnetic moments of the tau lepton in the processes e−γ→νeτν¯τ (γ is the Compton backscattering photon) and e−e+→e−γ⁎e+→νeτν¯τe+ (γ⁎ is the Weizsacker–Williams photon) with unpolarized and polarized electron beams at the CLIC. We have obtained 95% confidence level bounds on the anomalous magnetic and electric dipole moments for various values of the integrated luminosity and center-of-mass energy. Improved constraints of the anomalous magnetic and electric dipole moments have been obtained compared to the LEP sensitivity.