Resetting mid 2010 iMac to factory settings, but it will not let me install OS X El Capitan

The 27in Mid-2010 iMac shipped originally with OS X 10.6.3. You will need the gray DVD that shipped with the iMac to reinstall OS X 10.6.3, from which you can then perform software updates to OS X 10.6.8 v1.1.

If you no longer have the gray DVD that shipped with the iMac, then you will need to find a retail copy of the Snow Leopard 10.6.3 DVD on Amazon or eBay and boot from that DVD to get the operating system installed and update from there. I would apply all Snow Leopard updates through 10.6.8 v1.1 and the special Mac App Store update for it. Once you have done this, make a Time Machine backup of this operating system.

Now, using the Safari browser only, you may be able to obtain the full OS X El Capitan installer from the Mac App Store by accessing How to get old versions of macOS. The El Capitan download will drop into your Downloads folder as InstallMacOSX .dmg which you double-click to place the full installer into your /Applications folder. You can then now run that full installer to upgrade your iMac to El Capitan 10.11.6, and then apply any available updates via Software Update. I would make another Time Machine backup on a separate drive after the upgrade and updates are done for a baseline OS X El Capitan installation.

What to do before you sell, give away, or trade in your Mac This is what should be done be the previous owner ( Family Member ) of this computer and before you take ownership .