Provided are a computer program product, system, and method for sharing alias addresses among logical devices for a control unit managing access by hosts to logical devices configured with capacity from attached physical devices. An alias management group of logical devices and alias addresses assigned to the logical devices is configured. A plurality of requests to establish an association of the host with a logical device and the alias addresses assigned to the logical devices in the alias management group are received from a host. Acknowledgment is made to the host that the association is established in response to determining that the host is assigned the logical devices and alias addresses of the logical devices in the alias management group. The host can use one available alias address assigned to any one of the logical devices to access any one of the logical devices indicated in the association.