FWIW here are a few thoughts and comments:
1. A Normal tempate has to exist, Word will not run without one and it recreates a new Normal template if the original has been deleted.
2. Apple's Finder app stinks as a search tool. It does not search system folders unless you tell it specifically which one to search. Thus you cannot locate the Normal.dotm template with a general search of your system. The path to the template is: ~/Library/Application Support/Microsoft/Office/User Templates/Normal.dotm
3. If you have both a Normal.dot and Normal.dotm in the User Template folder, this will cause problems on exiting Word
4. If you have both Word 2008 and Word 2011 in the system, they both use the same Normal template from the same User Template folder could this cause the issue of the save message??
5. Certain versions of Norton Anti-Virus have caused this issue in the distant past on the PC ... could a current version be causing it again on the Mac???
6. Styles set in the document or the template to Auto Update could cause this issue.
7. Auto run macros in a global template that update something in the normal template could cause this issue.
8. Try starting Word in Safe Mode (hold down the Shift Key while starting Word) and do not have ANY other Office app running. If the behavior goes away it's probably a Global template in the Startup folder that's causing this issue.
9. Unless you have deleted files, renamed folders, or moved the Microsoft Office 2011 files and folders to other locations, reinstalling Office is probably not the answer.
10. A virus could casue this issue ... get a good program and scan your system.