Can I upgrade 2.4 GHz Intel Core 2 Duo, 2GB 667

YES YOU CAN!

Apple has determined that they do not care about older users, so chose not to include your machine for Sierra upgradability even though it is capable.

macOS Sierra requires at least 2GB of RAM and 8GB of storage space and will run on:

iMac: Late 2009 or newer.

MacBook and MacBook Retina: Late 2009 or newer.

MacBook Pro: Mid 2010 or newer.

MacBook Air: Late 2010 or newer.

Mac Mini: Mid 2010 or newer.

Mac Pro: Mid 2010 or newer.

But people like me, with a 2009 Mac Pro would not accept this. I upgraded mine.

Here's the post on it:

EFI and Firmware update from 4,1 to 5,1 for Sierra Upgrade Solved

Some more info you may want to go over before installing: https://forums.macrumors.com/threads/mac...

You should be able to do your's (note, I have not done this on your particular machine). Please make a backup of your current system prior to attempting this. You will need a copy of Sierra before you start, as your iMac is too old it won't let you download it so you will have to use another mac that is supported to do this. On the supported mac download Sierra, and before it starts the installation look in the Applications folder for Install macOSSierra.app and copy it to a USB stick, or send via network to your old iMac. I would advise installing it to an external drive, you should also check that any third party apps you are using are supported or have updates for Sierra, check here: https://roaringapps.com/apps?platform=os... also check whether any peripherals (printers, scanners, etc) are supported in Sierra, check with the manufacturer.

http://dosdude1.com/sierrapatch.html