For a second-order system with time delays and power nonlinearity of the degree higher than one, which does not contain a velocity-proportional damping term, the conditions of local asymptotic stability of the zero solution are proposed. The result is based on application of the Lyapunov-Razumikhin approach, and it is illustrated by simulations. Our local stability conditions for nonlinear systems are less restrictive than stability conditions of the corresponding linear models.