Microsoft Office 2016 Mac compatible with Big Sur 11.X?

The other possibility, albeit remote is to find out if Microsoft has a crossgrade policy for stand alone or Office 365 for Windows. If they do, you could put Windows 10 on your Mac, and maybe even Windows 11 if you have enough RAM and use Office on the Boot Camp or other virtualization tool.

Regardless, that's a Microsoft support question. You would also have to get a Windows license once you determine which version will run on your Mac's hardware.

Microsoft retired Office 2016 for Mac on Oct 13, 2020, and macOS Big Sur was released on Nov 12, 2020.

There is a very high likelihood that Office 2016 was never tested by Microsoft on Big Sur, and even though you can perform a Big Sur upgrade on your two Macs, there is no assurance that the currently installed and licensed Office 2016 applications will behave as expected afterward. Although either of your Macs can upgrade as far as macOS Monterey, that also increases the unknowns for Office 2016 for Mac functionality, and falls one operating system short of current Microsoft 365/Office 2021/2024 support for new installations.

You can get the full Big Sur installer from How to download and install macOS - Apple Support and I do recommend that you use Apple's Safari browser for that process.

Be certain to have a last Time Machine backup on both of your Macs and set that backup drive aside in case you need to revert the operating system should Office 2016 for Mac not work out for you after the upgrade.

Even if it is compatible, Microsoft in its infinite wisdom only has its licensing server send licenses to Macs running

Sequoia, Sonoma, and Ventura on fresh installs of Office.

This means in order to migrate Office to another Mac, you either using migration assistant on a stand alone license on an original Mac, or backup clone of said Mac, or deal with the Office incompatibilities of an Office clone or the web Office 365. These stand alone licenses of Office typically can only be run on one Mac at a time without one trying to home phone to Microsoft and limiting the access of the other Microsoft Office installation. Not to mention their license agreement state these restrictions.

Softmaker Office has licenses still available for older versions of MacOS and much better compatibility than any other Office clone, including the web Office 365. It is the only one that did not mess up the formatting of my résumé.

So unfortunately unless you have a running copy of Office on one of those older OSes, your options are limited for Office 2016.