<p>The nonlinear Moore–Penrose metric generalized inverse was introduced to analyze the best approximate solutions to ill-posed operator equations in Banach spaces. In this paper, under suitable conditions, utilizing certain geometric properties of Banach spaces and features of the metric projection, we develop some iterative methods for computing the Moore–Penrose metric generalized inverse of Banach space operators, deriving some conditions for iterative convergence. Furthermore, we provide error bounds for the iterative methods approximating the Moore–Penrose metric generalized inverse. As a supplementary component to our main findings, an iterative method for deriving the best approximate solution to ill-posed operator equations is proposed. Several relevant examples are also provided to illustrate the efficacy and application of these iterative methods. The results presented in this paper provide new insights into the analysis, computation, and applications of nonlinear generalized inverses.</p>