Nonlinear algebraic equations have wide applications in scientific computation and engineering application. In this paper, the inexact Newton-MCG algorithm for computing the reflexive solution of the nonlinear algebraic equation is proposed. The algorithm is constructed based on the Newton method for calculating the reflexive solution of the nonlinear algebraic equations and the modified conjugate gradient method for the approximate reflexive solution or the approximate reflexive least-square solution of the linear algebraic equation derived from each Newton step. Moreover, the proposed algorithm only requires the nonlinear algebraic equation to have the reflexive solution and the solution may not be unique, owing to the wide scope of applications and the finite-step convergent property of the MCG method. Finally, some numerical experiments illustrate the efficiency of the new algorithm.