2011 iMac stuck on Apple logo, unable to enter safe mode

There is only one way to boot an iMac (Intel processor) into safe mode:

1. Turn on or restart your iMac, then immediately press and hold the Shift key until you see the login window.
2. Log in to your Mac.
3. You might be asked to log in again.
4. On either the first or second login window, you should see “Safe Boot” in the menu bar.

If you are using a wireless keyboard, you may need to get your hands on a wired USB keyboard to trigger the safe boot. Wireless keyboards often don't respond to the computer quickly enough after restart.

Start up your Mac in safe mode - Apple Support

I think what you really need to do is boot macOS Recovery. Explained here:

If your Mac doesn't start up all the way - Apple Support

Verbose mode is a fine idea, but prior to that you reported it got stuck on the Apple logo:

If your Mac starts up to an Apple logo or progress bar - Apple Support

If your Mac is stuck on this screen or keeps restarting explains. Follow those numbered steps.

If that is as far as the boot process gets, then you have a major problem somewhere since the boot log hasn't even filled the screen.

First guess, a failing Hard Drive.

Second guess, bad GPU/Video card if it is a 27" model (it was known for GPU issues)

Third+, it could be anything including a corrupt OS, or other hardware issue not listed here.

Options:

• Try running the Apple Diagnostics
• Try booting into Internet Recovery Mode using Command + Option + R (assuming macOS 10.13 was installed at some point in the past....also use a wired USB keyboard since timing is critical). You can try repairing the file system, or reinstall macOS, or install macOS onto an external USB3 SSD & booting the iMac from the external.
• If Internet Recovery Mode is not accessible, you can create a bootable macOS 10.11 or 10.13 USB installer using another Mac from 2007 to 2018. (10.13 needs Late-2009 to 2018 Mac, while 10.11 needs a 2007 to 2015 Mac)
• Here is an Apple article with generalized instructions for various boot issues determined by the icon on the screen or system behavior: If your Mac doesn't start up all the way - Apple Support