iMac will not boot into recovery or internet recovery

Sorry to ask the obvious, but you have checked they actually have a recovery partition?

diskutil list

Should tell you pretty quickly.
No idea on the internet recovery, but potentially it requires the Recovery partition to be present as well...

Not all Mac's can do internet recovery.
Out of my head the 2007 models will never do that, the 2010 models require a firmware update.
I think that models that support internet recovery will have the the buttons to wifi access at the startup manager.

And re-imaging is normally done by netboot, not from recovery partition, so I would try to netboot and re-image the iMacs.

Create a boot installer USB stick. That will boot into the same thing as a recovery partition, if there is something you need to do in there.

The 2011 iMac should be able to do an Internet Recovery if it had Lion (OS X 10.7) or later, but if the recovery partition is missing then you have to recreate it &/or find another source for the installation.

If you have at least OS X 10.6.8 and a functional Apple ID, then you can download a copy of the OS X installer from the App Store and make use of this utility called Recovery Partition Creator:

Recovery Partition Creator

Edited previous post to clarify that the 2011 iMac should be able to do Internet recovery. The 2007 Macs will most likely not be able to do that, however.

Here is the list of systems that Apple says can be upgraded to include support for Internet Recovery. Anything newer than the machines on this list already includes support. Anything older is out of luck.

https://support.apple.com/en-us/HT202313

Thanks for the help guys. They do not have a recovery partition. The Faculty member that used to manage all of the macs on campus before i got here in January imaged them, how i do not know, but they have no recovery partition and are not recognizing an external drive with the installer on it... I will attempt the netboot method next. Thanks again guys!