checking for software update spin indefinitely

Have you tried rebooting the Mac?

If so, then try booting into Safe Mode to see if you can check for updates.

What version of macOS is your Mac currently running? And what is the exact model of your Mac? You can get this information by clicking the Apple menu and selecting "About This Mac".

Predicated on the user profile as being accurate " iMac 21.5″, macOS 11.6 "

There are several updates available for Big Sur as the current version has reached 11.7.8

There are factors:

1 - Like poor Wifi connection

2 - Slow wifi connection

3 - Too many other people downloading at the same time and Apple Severs are overloaded

4 - Software on the computer interfering with the download process including AntiVirus Software

Notation: Disabling the AV Software may or may not solve the issue. Removal of the AV Software may increase the chances of the download to perform as expected.

5 - Running behind a VPN which may increase Packet losses or corruption of the download 

6 - If via wifi too many Other devices using the wifi at the same time reducing the Bandwidth to your computer. 

7 - Best Results are via Ethernet Cable to Router

8 - Restart in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal and Try Again using the Software Update Applet in System Preferences