To achieve encirclement control of non-cooperative targets by leader agents, a cooperative encirclement strategy integrating prescribed-time convergence and prescribed performance constraints was proposed. A prescribed-time geometric center estimator was designed, which enabled the leader agents to accurately estimate the geometric center of the non-cooperative target agents within a user-specified time. Building upon this, combined with the designed velocity field and prescribed performance function, an encirclement control strategy was developed. This strategy not only drove the leader agents to form a circular formation for encirclement but also ensured that the geometric center of the non-cooperative target agents moved along the desired trajectory. Based on Lyapunov stability theory, it was rigorously proven that the proposed control strategy guaranteed that the encirclement error converges within the prescribed performance function and within a prescribed time. Numerical simulations validated the effectiveness of the proposed strategy. The results demonstrate that the proposed control strategy achieves precise encirclement control within a prescribed time and guides the geometric center to follow the desired trajectory. The proposed approach not only ensures that the upper bound of the convergence time is independent of the system′s initial state but also satisfies prescribed performance constraints, providing reference for the cooperative control of multi-agent systems.