iMac can't find external Time Machine backup drive

What make/model is that external drive and how old is it..?

If that external drive is not seen by Time Machine or by Disk Utility..

Then it has a faulty USB connection or has completely failed.

To test that external drive and possibly get it working again..

Test it plugged into a different port on the iMac.

Test it using a new or different USB cable.

Test it connected to a different Mac.

All hard drives fail, some will give you a warning however most do not. If your Time Machine External Hard Disk is 5 years old or older, there is a good chance that it has failed.

You are wise to use Time Machine however because hard disks fail many experienced users use redundant backup. What many (include me) do is use Time Machine for backup #1 and for a redundant backup they use a second External Hard Disk and use an app such as Carbon Copy Cloner. Carbon Copy Cloner sells in the $ 40 range however you would need to verify this. This is a good strategy because when 1 External Hard Disk backup dies then you have another you can use until you replace the failed drive and you have the peace of mind knowing your data is safe!

You can try restarting your iMac and then check in Disk Utility to see if your current Time Machine drive has failed or not. If it does not show up then replace it ASAP!!!!!!!!!!! If it shows up then attempting to use it for Time Machine and if that works then simply buy that second External Hard Disk and begin doing a clone backup as your redundant backup.

When it comes to EHDs they are NOT all the same, the "Walmart" quality drives tend to be a poor choice as the enclosures the manufactures use do not get adequate cooling and heat is what kills many types of electronic components. It is wise to spend a little bit more and buy high quality enclosures such as the OWC Mercury Elite Pro (https://eshop.macsales.com/shop/owc-mercury-elite-pro) . In the long run these will be less expensive as the HD's can be replaced easily and cheaply so you can re-use the enclosure.

Those external drives have a high failure rate.

Replace it with a better quality external drive and start new.Time Machine backups.

Back up your Mac with Time Machine - Apple Support

IMHO WD's enclosures tend to be pretty poor, the HD's inside are good but if the enclosure does not provide adequate cooling that shortens the life expectancy substantially . In addition, you mention its is probably over 5 years old which also isn't in your favor. Finally the light coming only means it is getting power, but if the HD has failed then it's dead.

I didn't know, what this means IMHO " In my Humble opinion", So I taught us a little today!

I never use my external Harddrives for more than 2 to 3 years, for my Time Machine Back-ups!

I also running, Monterey 12.7.6 on a Vintage MacBook Pro from 2015.

I have also used Western Digital External Harddrives, but now I used other Brands instead.

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