A 2011 iMac should be able to use any OS X version from 10.6.6 (Snow Leopard) to 10.13.6 (High Sierra). The more recent ones like High Sierra are available free from the App Store, but difficult if you don't have another OS X machine to do a download and make a bootable USB stick.

Update: I have tried a 10.9 image which didn't work as it is too recent, but am in the process of trying a 10.7 image. 

Also have tried lots of Virtual Machines to no avail. Thanks to everyone taking the time to help so far.

10.9 should not be too recent. Should be fine up to 10.13 https://everymac.com/systems/apple/imac/specs/imac-core-i5-2.5-21-inch-aluminum-mid-2011-thunderbolt-specs.html

Sorry I didn't clarify. To do a clean install on to a new HDD, the version of OS X/macOS cannot be too recent otherwise you get an 'O' with a line through it. My iMac can UPDATE to 10.13, which is what I think you mean, it just can't directly install a later version. Thanks anyway.

willyzz:

 

Update: I have tried a 10.9 image which didn't work as it is too recent, but am in the process of trying a 10.7 image. 

 

Also have tried lots of Virtual Machines to no avail. Thanks to everyone taking the time to help so far.

 

Do you mean you've been unable to run the VM's or the media created by them won't work?

I have installed High Sierra directly onto a iMac 2009 with a new hard drive (Previously had 10.7 installed) using the Dosdude1 patcher so you shouldn't have issues installing a newer version. The Firmware update for your machine should be part of the installer.

Media created by the VMs won't boot.

I have just tried a 10.7 .dmg USB which resulted in a blank white screen. Currently running another AHT to see if a video card or something is at fault too. I might have to check out the dosdude patcher.

Sounds like potentially an issue with the USB drive or the way you are creating it rather than a compatibility issue with the iMac

I will try another USB, and I am using TransMac for the media creation. Any suggestions please?

I've never used TransMac but depending on how the .dmg has been created it can be a but more complicated than just 'cloning' the image. After passing through the USB drive to your VM - I would use disk utility to format the disk "GUID / MacOS Extended" - then use https://macdaddy.io/install-disk-creator/ 

There is a program called 'createinstallmedia' that Disk Creator uses to turn the IntstallESD into the bootable USB - it can also be run from the terminal if you prefer to do it manually.

That sounds like a good route, however, I only have access to Windows machines, hence the use of TransMac...

I tried VMWare and VirtualBox macOS for a day or so and my Virtual OSes were getting errors or were stuck at the Apple logo screen.

Could try Rufus https://rufus.ie

I used rufus with the ignore bootable option (alt-m) and burned the snow leopard iso from archive.org to a usb drive. It showed up and was bootable on a 2013 MBP but kernel panicked cos newer mac.

Okay trying that now, will update soon. Thanks for continued interest in the problem. 

EDIT: Update:

Rufus Snow Leopard iso was actually recognised but resulted in Kernel Panic, and OS X 10.7.2/.4 resulted in white screen.