In this paper, the stabilization problem of nonlinear systems with time-varying input and state delays is considered by an approximate predictor. Due to the introduction of state delays, the nonlinear dynamic without state delays is used to predict the plant state, and a new error term is introduced into the approximation error between the prediction value and the future state of the plant. By Lyapunov-Krasovskii functional and small-gain argument, the stability of the whole closed-loop system is guaranteed provided that the sampling period, the predictor accuracy, and the upper bound of state delays satisfy an inequality constraint. The effectiveness of the proposed results is illustrated by a numerical example.