The paper proposes an approximate closure procedure for hierarchies of macroscopic equations for rarefied gases, derived as moment equations from the Boltzmann equation in kinetic theory of gases. The procedure is based upon application of the maximum entropy principle. If the exact minimizer is exploited, moments of the distribution function may diverge, unless the restriction on the structure of the moments is introduced. In this paper, a perturbative approach is proposed by restricting the set of admissible functions in the variational problem. This leads to an approximate minimizer, but the procedure can be applied to an arbitrary choice of the moments.