I was given a iMac (circa 2007, 12 Gig RAM, 1 Terabyte hard-drive, 27˝ screen) which when booted up suggested either reinstalling the operating system or reverting to an earlier save point.
All I wanted was to install Linux, and I decided on Linux Mint 21 Cinnamon

I followed the following steps, that seemed generally recommended:

1) Plug the bootable Linux USB drive into the Mac.
2) Turn on the Mac while holding down the Option key. You will then see the boot manager with a list of available devices you can boot from.*
3) Select your USB stick and hit enter. This will be named EFI boot or EFI drive.
4) You will see a screen with Installation Options that will ask you to either Try or Install.

When I tried, the screen simply went dark but after reading further, I tried again:

5) Press the letter ‘E’ on your keyboard to go to the boot entry.
6) I then followed the instructions to modify the boot entry: Change the line starting with Linux and add the word ‘nomodeset’ after ‘quiet splash’.

At this point Linux Mint ran quite happily as a live-disk, and I decided to install it, telling it to use the entire hard-drive.
The installation seemed to go correctly, until the end, where I was told I could restart the computer. It did not restart—instead it showed five lines of message:
[ 1447.698588] sd 1:0:0:0: [sdb] tag#0 access beyond end of device
[ 1447.698646] blk_update_request: I/O error, dev sdb, sector 4783632 op 0x0 : (READ) flags on 80700 phys_seg 1 prio class 0
[ 1447.699993] sd 1:0:0:0: [sdb] tag#0 access beyond end of device
[ 1447.700021] blk_update_request: I/O error, dev sdb, sector 4783632 op 0x0 : (READ) flags 0x0 phys_seg 1 prio class 0
[ 1447.700064] Buffer I/O error on dev sdb2, logical block 1040, async page read

I attempted the installation a second time, with exactly the same results.
I then tried again, with a flash-drive carrying MX Linux. Unfortunately, at step 5, although pressing ‘E’ opened a terminal, there was no boot entry showing.

Wondering if the five lines of message suggested a problem with the hard drive, I ran the live disk again, and used Gparted to divide the main partition into two, and then tried to install Mint to the first of them, but received the same message. I then tried to install to the second partition, with the same result.

Note that each time I have attempted an installation (after the first time) when asked where I want Mint installed, the live disk has said that Mint is already installed.

Would another distribution have better luck, or is there some other solution?
Thanks!

You could try with another hard-drive (external or not) to try to to determine if your hard-drive may be faulty.
Did you have macOS running on this hard-drive before?

I would retry MX leaving it on auto (without editing "boot").

1 members found this post helpful.

I decided to try MX again (although I did not understand 'leaving it on auto').
When the live-disk opened it showed five lines:

MX 21.3x64
Language-Keyboard-Time
>>>Advanced Options<<<
Memory Test (64 bit)
Boot Rescue Menus

Seeing it a second time, I realized that it had defaulted to Language-Keyboard-Time, which I thought explained my lack of success with 'e' the first time around. I chose the first option, MX 21.3, and then 'e' and this time the terminal showed information, the last line of which ended in 'quiet splasht no splash'. I added nomodeset, and the installation proceeded without any difficulty.
It did show a message:
'The disks with the partition pass the SMART monitor test ... but will have a higher than average failure rate in the near future.'
Considering the age of the computer, hardly a surprise.
Very satisfying! Thanks!