Existing CoAP primarily focuses on communication (i.e., client/server-based request/response interaction) between constrained devices in a constrained network. Constrained networks may have large latencies and constrained devices may not be able to respond in a timely manner. Therefore, incorporating delay awareness in the CoAP protocol may prove beneficial for communications between devices. This article presents a new method to support a delay-aware CoAP messaging model, where a Delay- Indicator is proposed to be contained in messages to reflect delay tolerance information. Furthermore, the Delay-Indicator can be leveraged by CoAP Clients and CoAP Servers to improve their operations of entering sleep mode to save energy while not violating delay tolerances. With this new enabler, a timeout mechanism for CoAP Responses is put forward to enhance CoAP performance. Simulation results show that incorporating delay awareness in CoAP significantly improves messaging performance.