The asymptotic equivalence of between completely positive Toeplitz maps and completely positive circulant maps is proved. By analyzing block diagonal matrices with Toeplitz blocks resulting from the restriction to some invariant subspaces, an ad hoc asymptotic equivalent sequence of circulant CP maps is constructed under the hypothesis of the generating symbol being of the Wiener class. The underlying structure of the circulant maps is a block diagonal with circulant blocks.