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 m(a) 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 m(a) = 1000 - 2000 GeV with collision energy of 3000 GeV and integrated luminosity of 4000 fb(-1) 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 m a > 80 GeV. In particular, they are more restrictive than the limits that follow from the ALP-mediated LBL scattering at the LHC.