Which SMB protocol is used by Time Machine?

Big Sur only uses SMB, now.

In order for Time Machine to use a NAS as a backup destination, the NAS has to support the Time Machine over SMB specification. I doubt that a router that is only using SMB1 would be capable of supporting the Time Machine over SMB specification.

Even if it did, it is not a supported backup disk for Time Machine.

I have attached a Lacie 2TB hard drive to my Linksys router Model WRT3200ACM, intending to use it as a backup disk for my MacBook Air (Retina, 13-inch, 2020) using MacOS Big Sur Version 11.1.

That configuration is not supported by Apple. Good luck.

when I go to Time Machine preferences to choose this external drive as my location for Time Machine backups, this preference window does not show this disk as an option.

You would have to trick Time Machine that is a supported backup scheme.

I was told by Linksys support that this router model only supports the SMB1 protocol. Can anyone tell me which SMB protocol is used by Time Machine because I suspect this is why the Time Machine preference window does not recognize this external disk.

I think it can connect to any version, but it defaults to only File Sharing on 2 or 3. That would not affect connecting to your router. Just try to connect to the drive in the Finder. If you can connect in the Finder, then the version is not the issue.

If a router or NAS does not specifically say that it supports Time Machine,

it is a coin toss whether it will work and worse yet whether it will be

reliable, which is the whole point of backups. A backup that is not reliable

is no backup.