Let \(Q\) be a square circulant matrix whose row and column sums are zero. The problem of solving the regularized linear system \(Ax=b\) is considered, where \(A=Q^T+\epsilon I\), with \(\epsilon > 0\). Let \(D\) be the diagonal matrix whose diagonal entries are the corresponding diagonal entries of \(A\) and whose lower and upper parts give rise to the lower triangular matrix \(L\) and the upper triangular matrix \(U\), respectively. Here, the diagonal entries of \(L\) and \(U\) are all equal \(\epsilon /2\). Then \(A=T+S\), where \(T\) is a triangular matrix with nonnegative diagonal elements and \(S\) is a symmetric matrix with positive diagonal entries and negative off-diagonal elements. Such a splitting is referred to as a \(TS\)-splitting. It is shown that \(A\) is positive definite. An iterative scheme is proposed using the \(TS\)-splitting. It is shown that this scheme converges to the unique solution of the regularized system. Comparisons are made with the existing methods for a numerical example.