This paper investigates the distributed leader-following consensus control problem for nonlinear multi-agent systems with external disturbances under aperiodic intermittent communication. For the considered nonlinear multi-agent systems (MASs) with bounded disturbances satisfying Lipschitz continuity conditions, a distributed extended state observer (ESO) is designed to estimate and compensate the system states and aggregated disturbances in real time. Based on the characteristics of aperiodic intermittent communication, a distributed consensus control protocol with disturbance compensation capability is proposed. The protocol updates control signals only during communication phases while maintaining zero input otherwise, thus effectively reducing the communication burden and energy consumption of the system while ensuring system performance. By leveraging the Lyapunov theory and matrix inequality-related knowledge, it is proven that the proposed control protocol ensures the system can achieve leader-follower bounded consensus under complex conditions of aperiodic intermittent communication and unknown external disturbances. Numerical simulations verify the effectiveness of the presented control protocol.