The anomalous quartic neutral couplings of the γγγγ vertex in a polarized light-by-light scattering of the Compton backscattered photons at the CLIC are examined. Both differential and total cross sections are calculated for e + e − collision energies 1500 GeV and 3000 GeV. The helicity of the initial electron beams is taken to be ± 0.8. The unpolarized and SM cross sections for the same values of helicities are also estimated. The 95% C.L. exclusion limits on two anomalous photon couplings ζ 1 and ζ 2 are calculated. The best bounds on these couplings are found to be 4.58×10 −16 GeV −4 and 9.37×10 −16 GeV −4 , respectively. The results are compared with the exclusion bounds obtained previously for the LHC. It is shown that the light-by-light scattering at the CLIC has a greater potential to search for the anomalous quartic neutral couplings of the γγγγ vertex.