<p>This paper attempts to provide two co-design schemes that integrate event-triggered mechanisms and time-varying linear feedback for realizing the prescribed-time leader-follower consensus under a directed communication topology, where both state- and observer-based output feedback are considered. These designed protocols contain parameters related to the information of the directed graph and a time-varying parameter that corresponds to the parameter Lyapunov equations. To fully utilize limited bandwidth resources, event-triggered mechanisms are developed to coordinate the designed feedback using these time-varying parameters, where the unbounded time-varying parameters guarantee the prescribed-time convergence. Zeno behavior is avoided with an ensured minimal inter-event time, which is independent of the system itself and can be assigned arbitrarily with the time-varying parameter in some sense. The effectiveness of the co-design strategies is verified through two numerical examples.</p>