In this research paper, we introduce a novel gradient-free modified three-term conjugate gradient method designed to solve nonlinear equations subject to convex constraints. Our approach incorporates the projection scheme, which enhances the effectiveness of the proposed method. Building upon the modified three-term conjugate gradient method for solving M-tensor systems and l1- norm-based nonsmooth optimization problems, our method can be regarded as an extension of their technique. By making mild assumptions, we establish the theoretical convergence properties of our iterative method. Through extensive numerical experiments, we demonstrate that our proposed approach is not only highly efficient but also outperforms other existing methods in terms of performance and accuracy. © 2023 Institute of Advanced Engineering and Science. All rights reserved.