Recurring challenges in bilevel optimization involve the need to solve the lower level optimization exactly and the fact that first- and second-order methods require differentiability of its value function. In this paper, we propose a “gradient-free” framework to analyze inexactness in bilevel nonlinear optimization in the sense that it does not depend on gradients of the lower level optimization problem. To that extent, we derive a joint system of Karush–Kuhn–Tucker necessary conditions. We then solve the bilevel optimization problem by a multi-step Newton-type method for generalized equations. In this setting, we derive input-to-state stability properties for the bilevel optimization algorithms under moderate regularity conditions and provide sufficient conditions for convergence even if the lower level problem is solved inexactly.