A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradientconstraints in function space is considered. Problems of this type arise, for instance, in themathematical description of the magnetization of superconductors, in problems in elastoplasticity, orin electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs bya sequential minimization technique relying on the repeated solution of variational inequalitytypeproblems. A monotone operator theoretic approach is developed which does not resort to Moscoconvergence as is often done in connection with existence analysis for QVIs. For the numerical solutionof the QVIs a penalty approach combined with a semismooth Newton iteration is proposed.The paper ends with a report on numerical tests involving the p-Laplace operator and various typesof nonlinear constraints.