Some additional info... I found the Updates button in the Home panel. On one computer, I can start the Update but then it comes up with a notice that the update failed. On the other it has a notice that the update is in progress but it never seems to end.
On both computers I found the History button in the Mac Security panel. On one it shows the Automatic Update 20 times today. Double clicking on any of the lines shows "Update Failed", or "Another session running" or Downloading DAT" or several other messages. And still the New Version notice keeps coming up on both computers.