In this study, light-by-light (LBL) scattering with initial polarized Compton backscattered photons at the CLIC, induced by axion-like particles (ALPs), is investigated. The total cross sections are calculated assuming CP-even coupling of the pseudoscalar ALP to photons. The 95% C.L. exclusion region for the ALP mass and its coupling constant f is presented. The results are compared with CLIC bounds previously obtained for the unpolarized case. It is shown that the bounds on f for the polarized beams in the region with collision energy of 3000 GeV and integrated luminosity of 4000 fb are on average 1.5 times stronger than the bounds for the unpolarized beams. Moreover, our CLIC bounds are stronger than those for all current exclusion regions for GeV. In particular, they are more restrictive than the limits that follow from the ALP-mediated LBL scattering at the LHC.