IOS – iPhone 5 won’t show iOS11 update

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?

The next solution to try might be to backup/restore your iPhone.

The Apple Store can help you with this, but probably not free of charge unless your iPhone is under warranty.

Fortunately, you can do it yourself, but you'll need to be able to log into iTunes with the same Apple ID that is used to identity you on the iPhone on a computer, and you'll need to be able to connect to that computer using a USB adapter cable.

If you can do that, then here's a pretty good description of the process:

http://www.uberphones.com/2011/05/restore-your-iphone-and-keep-homescreen-organized/.