Yes, it is possible, but not easy. I would do this by putting an AutoNew in your macro-enabled template that changes the File Open directory (which is the default File Save directory).
The following macro will change the file save path and force an immediate save when a new document is created based on the template that contains the macro.
Code:
Sub AutoNew()
'
Dim strNewPath As String
strNewPath = "C:\desired location\" ' Set save location
ChangeFileOpenDirectory strNewPath
ActiveDocument.Save
End Sub
You would set the actual path in the line setting the value for strNewPath. Note that the path has to end with a backslash \
Users will need to have macros trusted in the template location. Note that this will temporarily set the place for File>Open if they try opening another document, just as it would if they were to manually save the file in that directory.