How do I re-use external hard drive from old iMac Time Machine backup to backup new iMac

if you want to keep the catalina TM, then you should buy a new EHD to use with big sur TM. the drives are readily available, and their price seems to get lower every month. and, i second rkaufmann87's advice on redundant backups. and also on replacing EHDs every 5 or 6 years.

Norway 4 wrote:

Do I need to wipe the external hard drive to make room?

when i installed big sur, that's exactly what i did. i ran big sur for a few days to first make sure i didn't want to revert to catalina. once i was comfortable, i completely wiped the EHD and ran a big sur time machine on the "fresh" drive. but, as a footnote, i only do a backup once per week. i know that's not ideal, but my mac is for entertainment, not work. and also i use a bunch of external storage drives so losing data on my internal drive, in my application, is not really something i fear.

You can do that however it is VERY wise to replace EHD HD's every 5-6 years as they will fail. The trick is to replace before a failure occurs, with HD;s it is not a mater of If a HD will fail, it is a matter of when. The trick is they are not predicatable. Believe me, you dont want to learn a HD has failed when you need it most. This why redundant backups are smart!!!!!