Circulant preconditioning for symmetric Toeplitz systems has been well developed over the past few decades. For a large class of such systems, descriptive bounds on convergence for the conjugate gradient method can be obtained. For (real) nonsymmetric Toeplitz systems, much work had been focused on normalising the original systems until Pestana and Wathen (Siam J. Matrix Anal. Appl. 36(1):273–288 2015) recently showed that theoretic guarantees on convergence for the minimal residual method can be established via the simple use of reordering. The authors further proved that a suitable absolute value circulant preconditioner can be used to ensure rapid convergence. In this paper, we show that the related ideas can also be applied to the systems defined by analytic functions of (real) nonsymmetric Toeplitz matrices. For the systems defined by analytic functions of complex Toeplitz matrices, we also show that certain circulant preconditioners are effective. Numerical examples with the conjugate gradient method and the minimal residual method are given to support our theoretical results.