IOS – Change iCloud Backup from new version of iOS to older version

Not all versions of iOS can back up from backups made using more recent versions (i.e. the backups are not necessarily forwards-compatible). Confusingly, rather than show these unusable backups in some way (e.g. disabled with an error message available on tap) they simply don't show up at all (at least with iOS 5.0.1).

A new-from-the-box iPhone isn't necessarily running the latest version of iOS; it's running what was latest (or close) when it was boxed. If the version on the phone is older than the version that was used to make the backup (which is actually quite likely now that iOS updates can be done over-the-air).

Thankfully, there is a simple fix: choose to set up the phone as a new phone, rather than from a backup. Skip the step where you are asked for an Apple ID, since you won't actually be using the phone like this.

Once the initial setup is done, plug the phone in to iTunes and choose "Restore" (you can skip the backup of the clean-state phone). This will download the latest version of iOS and install that first, and then offer to restore (either from backups on the computer or iCloud).

You may be stuck on iOS 7 if you don't have a way to get that software from Apple's servers. Typically only registered developers can download several versions of software, but if you search - there are many places that archive these software ipsw files. It's always a bit of a risk downloading software from a place other than the vendor, but I don't know of a better option.

Typically, when a new OS is released, iTunes will only download the latest software if you don't already have an older version of the software stored locally. Is there any chance that computer (or another computer) downloaded iOS 6 and still has it on the hard drive? If so, you can decline the offer to upgrade and restore the device from that version of firmware.

As was covered in the comments - there is no restriction to load iOS 6 backup on iOS 7. Also - this is more about the computer and iTunes having the old iOS file already downloaded than something that's inherent on the device. Any change you have a backup copy of the iOS files iTunes downloaded?