Abstract: This paper investigates the prescribed-time consensus problem for high-order nonlinear multi-agent systems. For the leader-follower multi-agent systems under directed graphs, considering that the dynamics of the agents are in strict feedback form and contain nonlinear terms satisfying time-varying Lipschitz growth rates, we propose a prescribed-time consensus control method. Firstly, we construct a time-varying high-gain observer to estimate the unknown or unmeasurable state information. Then, a feedback controller with time-varying gains based on the observer information is designed. By employing appropriate state transformations and error variables, the prescribed-time leader-following consensus problem of the original system is transformed into the prescribed-time stability problem of the following error system. The convergence performance is analyzed by Lyapunov stability theory, and the boundedness of the system state and the control law is further demonstrated. Finally, we carry out a simulation case to demonstrate the validity of the theoretical results.