The main purpose of this paper is consider Newton-Taylor polynomial solutions method in numerical solution of nonlinear system of differential equations. We apply Newton's method to linearize it. We found Taylor polynomial solution of the linear form. Sufficient conditions for convergence of the numerical method are given and their applicability is illustrated with some examples.In numerical examples we give two benchmark sample problems and compare the proposed method by the famous Runge-Kutta fourth-order method. These sample problems practically show some advantages of the Newton-Taylor polynomial solutions method.