Ah yeah, I've recently received for free an iMac running Macos 10.9. It's simply impossible to upgrade; the only proposed upgrade release is 10.11, the installation starts then fails in a loop. Fortunately I don't actually need to save anything from this machine, and I have another Mac to download a newer OS installer, but that's quite painful.
If it's a 2007/2008 model iMac then it will be able to run 10.11 (El Capitan). If it's a Late 2009 iMac or newer then it will be able to run at least 10.13 (High Sierra).
If the default/upgrade installation is failing then I'd try creating a bootable installer on USB [1]. If it still fails then try erasing the target drive first to do a clean install (you can do this by running Disk Utility from within the installer).
It's a 2014 model, it can definitely run Macos 11. But as it has been unused for the past few years, it hasn't been upgraded and it's quite funny how utterly unusable it became: very few websites work at all (certificates problem), it's impossible to install any current application because even Firefox LTS requires 10.13 or so, and it's impossible to upgrade without using another Mac to download the update tool. That's not very user-friendly if you ask me :)
