In real-time medical monitoring systems, given the significance of medical data and disease symptoms, a secure and always-on connection with the medical centre over the public channels is essential. To this end, an edge-enabled Internet of Medical Things (IoMT) scheme is designed to improve flexibility and scalability of the network and provide seamless connectivity with minimum latency. The entities involved in such network are vulnerable to various attacks and can potentially be compromised. To address this issue, an authentication scheme comprised of digital signature and Authenticated Key Exchange (AKE) protocol is proposed which guarantees only authorized entities get access to the services available in the medical system. Moreover, to fulfill the privacy-preserving, each entity is mapped to a different pseudo-identity. The non-mathematical and performance analysis show that the proposed scheme is robust against various attacks such as impersonation and replay attacks.