  We investigate the Wasserstein distance between the empirical spectral
distribution of non-Hermitian random matrices and the Circular Law. For general
entry distributions, we obtain a nearly optimal rate of convergence in
1-Wasserstein distance of order $n^{-1/2+\epsilon}$ and we prove that the
optimal rate $n^{-1/2}$ is attained by Ginibre matrices. This shows that the
expected transport cost of complex eigenvalues to the uniform measure on the
unit disk decays faster compared to that of i.i.d. points, which is known to
include a logarithmic factor.
