Summary: We consider solving ill-conditioned linear systems under nonnegativity constraints with noisy right hand sides. The classical approaches to solve such systems are constrained least squares (quadratic programming) and barrier methods. First we present these classical methods. Then a modified version of the GMRES method (NGMRES) is presented. Since we assume that the coefficient matrices are ill-conditioned, then the Tikhonov regularization of the problem is considered. Our computational experiments show that the NGMRES provides us high quality solutions much faster than the other two approaches.