This paper studies circulant approximations of Hermitian Toeplitz matrices that are solutions of certain minimization problems relative to the Frobenius norm, the $lub_1 $, or the $lub_\infty $ norms. The examinations supplement the family of the so-called optimal and superoptimal preconditioners that have been proposed for the preconditioned conjugate gradient method for solving Toeplitz systems $T_n x = b$. For the new Frobenius norm approximations it is shown that they can be computed in $O(n\log (n))$ arithmetic operations, and that the eigenvalues of the preconditioned linear equations are asymptotically clustered around 1 if $T_n $ is a leading principal submatrix of an infinite Toeplitz matrix connected with a positive function of the Wiener class.