A method performed by a wireless device for handling communication of data between the wireless device and a network node serving a cell of a wireless network, the method comprising: - receiving from the network node when the wireless device is in idle or inactive mode, system information related to availability of a potential data bearer dedicated for data of a potential data class which indicates that a certain delay for delivering the data can be tolerated, - when the data is downlink data initiated by the network, receiving a paging message from a core node associated with the wireless network, the paging message indicating that the paging is caused by pending downlink data of the potential data class, - determining whether the potential data bearer is available for communication of the data or not, based on the received system information, - transmitting to the network node, when the potential data bearer is available for communication of the data, a message of a service request procedure that initiates a transition from the idle or inactive mode to a connected mode, and communicating the data with the network node over the potential data bearer when the wireless device is in the connected mode, or - staying in the idle or inactive mode when the potential data bearer is not available for communication of the data, wherein said staying in idle or inactive mode when the potential data bearer is not available comprises omitting a response to the paging message or sending a page response message to the core node with an indication that the potential data bearer is not available, wherein the page response message comprises a paging periodicity timer instructing the core node to wait with further paging to the wireless device caused by said pending downlink data until the paging periodicity timer has expired. - receiving from the network node when the wireless device is in idle or inactive mode, system information related to availability of a potential data bearer dedicated for data of a potential data class which indicates that a certain delay for delivering the data can be tolerated, - when the data is downlink data initiated by the network, receiving a paging message from a core node associated with the wireless network, the paging message indicating that the paging is caused by pending downlink data of the potential data class, - determining whether the potential data bearer is available for communication of the data or not, based on the received system information, - transmitting to the network node, when the potential data bearer is available for communication of the data, a message of a service request procedure that initiates a transition from the idle or inactive mode to a connected mode, and communicating the data with the network node over the potential data bearer when the wireless device is in the connected mode, or - staying in the idle or inactive mode when the potential data bearer is not available for communication of the data, wherein said staying in idle or inactive mode when the potential data bearer is not available comprises omitting a response to the paging message or sending a page response message to the core node with an indication that the potential data bearer is not available, wherein the page response message comprises a paging periodicity timer instructing the core node to wait with further paging to the wireless device caused by said pending downlink data until the paging periodicity timer has expired.