The generalized minimal residual (GMRES) method equipped with a stopping criterion based on the discrepancy principle is used to solve nonsymmetric nonsingular ill-posed problems with a right-hand side contaminated with errors. When the inverse operator \(A^{-1}\) of the linear system \(Ax=b\) is not bounded and the exact right-hand side \(b\) is not known, but \(b^\delta\) is, then a new linear system \(A_{\text{reg}}^\delta x = b^\delta\) is solved, where \(A_{\text{reg}}^\delta\) has bounded inverse. The authors propose to obtain \(A_{\text{reg}}^\delta\) by applying the GMRES method until \(\|b^\delta - A_{\text{reg}}^\delta x_{\text{reg}}^\delta\|\leq \alpha \delta\), for some fixed \(\alpha >0\), and it is proved that if the GMRES method solves the original equation in finitely many steps, then the GMRES equipped with the proposed stopping rule is a regularization method. Numerical experiments are done and the method is compared with the conjugate gradient least squares method showing good performance.