iMac keeps restarting without booting up

Thank you for your response!! I had tried the Shift key before and it didn't do anything. Tried it again, but nothing changed. Holding the Option key down did give me a choice of which mount to use - the standard disc or the Recovery disc. I chose "Recovery" and it started to boot, BUT it stopped at a grey screen and is now stuck there. Tried again and chose my standard disc, and the same thing happened. I can also choose a WiFi network to join, but that didn't make a difference either. Next - I'll see if I can connect my Macbook to this iMac and try to boot it from there. Someone asked me to try that. Any experience with that method?

I connected my Macbook to the iMac using a Thunderbolt cable, booted the Macbook with "t" pressed, and then started the iMac with "alt/option" pressed. The Macbook's disc is not showing up as a boot-up option. However, when I restarted the iMac while holding down the "t" key and the Macbook while pressing "alt/option", the iMac's hard drive showed up in the Macbook. I didn't boot from there because the iMac is where the problem is and not the Macbook, However, I was at able to take a data backup onto an external drive. So that was good.

But I'm still not able to boot the iMac from the Macbook.

BDAqua - first of all - thank you for your responses. They seem to help me get closer to the elusive goal. I was able to successfully mount the iMac drive on my Macbook through the Disc Utility and run a first aid repair. It ran successfully - but when I tried rebooting the iMac it still had the same behavior - booting up to a grey screen and then automatically rebooting again and again. Macbook OS is Mojave 10.14.6. I don't remember what the iMac was on, but it was most likely the most updated version as of early 2020.

I restarted the iMac today in regular mode and miraculously it booted up!! Perhaps the disk repair worked but needed some kind of rest? I don't know. But looks like we're back in business. Thank you so much BDAqua!!!