For a targeted wireless device that is operating in a communication network that includes a first access network and an associated core network, the teachings herein advantageously provide for the conditional selection of a second access network, for exchanging NAS signaling between the communication network and a targeted wireless device, to offload such signaling from the first access network. The second access network is different from the first access network and may or may not be integrated with the core network, but the first access network and/or the associated core network are adapted to provide for the conditional routing of certain NAS signaling through the second access network, which NAS signaling would conventionally be conveyed via the first access network. In an example, the first access network is a 3GPP RAN and the core network is a 3GPP core network, while and the second access network is a non-3GPP RAN.