Verdict: Not directly
Not directly. Blizzard make no Linux version of the Battle.net app, but it installs and runs on Linux through Lutris, which has a ready-made setup for it.
Battle.net is a Windows program. Blizzard do not publish a Linux build, and their games are sold with Windows and Mac requirements only.
The usual route is Lutris, a free launcher that installs Windows programs on Linux using Wine, a translation layer that converts Windows instructions as they run. Lutris has a community-written setup for Battle.net, so you pick it from a list, sign in with your normal account and your games install as they would on Windows. Bottles does much the same job with a different approach.
Your account, purchases and saved characters are untouched. They live on Blizzard's servers, so signing in from Linux brings everything back.
Be realistic about what this is. It is an unofficial arrangement that Blizzard do not support, and when an update breaks it you wait for the community to work out a fix. Games with strict anti-cheat systems can also refuse to run even when the launcher itself is fine, so check your specific game before you commit.
The usual way people run Battle.net on Linux. It has a prepared setup, so you pick Battle.net from a list rather than working it out yourself.
Your files: Your Blizzard account, purchases and cloud saves are unaffected. Installed game files are downloaded again.
Keeps each Windows program in its own tidy space. A cleaner way to do the same thing if Lutris feels cluttered.
Your files: Same as above. Nothing on your Blizzard account changes.
Not a replacement for Battle.net, but the store where Linux gaming works best. Some Blizzard titles are sold there.
Your files: A separate store with separate purchases. Blizzard games bought on Battle.net do not move across.
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