We investigate cosmological solutions of the chameleon model with a non-minimal coupling between the matter and the scalar field through a conformal factor with gravitational strength. By considering the spatially flat FLRW metric and the matter density as a non-relativistic perfect fluid, we focus on the matter-dominated phase and the late-time accelerated-phase of the universe. In this regard, we manipulate and scrutinize the related field equations for the density parameters of the matter and the scalar fields with respect to the e-folding. Since the scalar field fluctuations depend on the background and the field equations become highly non-linear, we probe and derive the governing equations in the context of various cases of the relation between the kinetic and potential energies of the chameleon scalar field, or indeed, for some specific cases of the scalar field equation of state parameter. Thereupon, we schematically plot those density parameters for two different values of the chameleon non-minimal coupling parameter, and discuss the results. In the both considered phases, we specify that, when the kinetic energy of the chameleon scalar field is much less than its potential energy (i.e., when the scalar field equation of state parameter is $\simeq - 1 $), the behavior of the chameleon model is similar to the $\Lambda CDM$ model. Such compatibility suggests that the chameleon model is phenomenologically viable and can be tested with the observational data.
