While there is currently no protocol on transferring Whisper accounts between phones, some users have had success trying the following method on iOS devices:
Make sure you recently backed up your iPhone to your iCloud account. A complete backup of your phone's current state is needed in order to transfer a Whisper account to a new device.
Follow Apples guide to back and up and restore a device via iCloud.
Once you have backed up your device’s information to iCloud, you must restore the new device from an iCloud backup. If you set up your new phone as a new device and download Whisper your account will not transfer.