AirPods Pro 3 not connecting to iMac running macOS Big Sur

Curtis,

We don't have any information on what may be causing your problem, so first do a Safe Mode restart. If you are using an Apple wireless keyboard, please connect the charging cable for the keyboard to the back of the computer and the keyboard. This makes the keyboard a "wired" keyboard and ensures the Safe Mode restart works properly. Continue to hold down the Shift Key on the restart until the computer starts up, then restart the computer normally and re-try your AirPods Pro connection.

If you continue to have problems then give AppleCare a phone call, you can find their phone number for your area by clicking AppleCare Contact Info.

NOTE: You have a VERY old iMac and a VERY old version of Mac OS. While your AirPods Pro 3 should work you may also want to begin considering replacing the old iMac with something newer. Why, simple....when you use very old technology eventually you will run into compatibility issues with new technology (both hardware and software). As a general rule of thumb it is wise to replace your Macs about every 6-8 years so you don't run into these issues.

If you STILL have a problem, please do the following.

It will help us get a good idea of what is causing your problem if you provide a report of your system, that way we do not have to play 20 questions with you. The report we are requesting does NOT provide any personal information and is extremely safe to use. Please navigate to www.Etrecheck.com and download the free version of EtreCheck. Once you have you downloaded the app and installed it, please run the report and save it. This report will help us get a good idea what has been installed on your system and help us be able to diagnose what may be wrong.

When you have your report, you can attach it when you reply to this message and we can then review it and help you determine what is needed to get your system running well again. 

For instructions on how to download your EtreCheck report and attach it to your reply to this message please click How to use the Add Text Feature When Post… - Apple Community It is helpful to us if you download the report to your Desktop.

Those two Apple Support document are a good start. For what it's worth my AirPods Pro 3 are working fine with this Mac running Big Sur, so it stands to reason yours should also.

Try resetting the Mac's BT module by pressing and holding ⌥ ⇧ while clicking the BT icon in the Mac's menu bar. Three previously unseen options will appear: reset, factory reset, and remove all devices. Obviously you won't want to choose any of them unless you already have wired input devices available (keyboard and mouse). You can use any generic ones.

You've done just about everything I've seen recommended.

I'd take it in to the nearest  Apple Genius Bar or Apple Authorized Service Provider and have them work on it.

Your Mac is obsolete and macOS Big Sur is no longer supported.

Many features of the AirPods Pro 3 require macOS Sequoia or later.

I'm not surprised you're having difficulty maintaining a connection to your old Mac. There is a limit to backwards compatibility of bluetooth. You seem to have reached it.