In this article we study the convergence of the positions of a multi-agent system in a cyclic pursuit under asynchronism and time delays. Each agent is assumed to operate on an infinite sequence of behaviors modeled by a finite state machine, which is represented by a discrete asynchronous mathematical model on a higher-level. The results on the convergence of the synchronous model are used in the proof of convergence of the asynchronous system. Numerical simulations are also performed to verify the theoretical results.