iMac27" late 2013 running Catalina. Can it be used as a monitor with an M1 Mac Mini?

The old-style Thunderbolt ports on your 27" Late 2013 iMac have Mini DisplayPort connectors, and can

• Provide Thunderbolt signals for a Thunderbolt dock or a 27" Apple Thunderbolt Display
• Provide DisplayPort signals for a Mini DisplayPort display or adapter
• Accept Thunderbolt signals for Target Display Mode. (No DisplayPort signals need apply.)

There is no dedicated "TDM socket" on your iMac or on any other iMac that supports Target Display Mode. On all such iMacs, you simply plug the external video source into one of the mDP or TB sockets normally used for output.

However, your M1 Mac mini does not support using any iMac as a Target Display. In addition, if your iMac is having hardware problems that cause it to crash or hang at arbitrary times, there's no guarantee that it would be usable as a Target Display even for another old Mac.

Time to get a real hardware monitor.

sapper wrote:

My apologies, but when I asked the question as to whether a Mac Mini could be used with my iMac, I was merely thinking it COULD be done. I don't actually own a Mac Mini....yet.

Sorry, my mistake. Still, Apple's current restrictions on Target Display Mode make it all but useless.

The only Mac minis – or other computers – now supported as video sources are Macs released in 2019 or earlier, running Catalina or earlier. No new Macs, and no Apple Silicon Macs of any sort, can run versions of macOS that old. You'd be buying an old Intel-based Mac so that you could remain stuck on Catalina, while using the old iMac display in a way that is less convenient than the way you are using it now.

If the "new" Mac had Thunderbolt 3 ports, making a Thunderbolt connection would cost you another $80+ (for an Apple Thunderbolt 3-to-2 adapter and a Thunderbolt 1/2 cable).

There are easier ways to flush your money down the drain and to torture yourself …

(Reference: Use your iMac as a display with target display mode - Apple Support)

No, it's not possible.

No: Using an iMac as a monitor - Apple Community

I have had a notification advising me to use a different browser other than Safari.

Regard all those notifications as suspicious. I don't use Facebook or anything like it but such messages can be fraudulent, misleading, or simply wrong. In the event it is legitimate consider using Brave and / or Firefox, both of which will be newer than Catalina's version of Safari.

Nope! The M1, M2 or M3 Mac mini's can not use your 2013 iMac as a cabled display in Target Display Mode.

For a little help deciding on your next iMac or Mac mini model.

see > Mac - Which Mac is best for me? - Apple

The Mac mini's are awesome little "bring your own display(s)" machines.

see > Connect a display to Mac mini - Apple Support

I recommend a new M4 or refurbished M2 directly from Apple, instead of used one from joe blow.

see > Buy Mac mini - Apple and > Refurbished Mac - Mac mini - Apple