Thanks for your advice. I haven't got around to looking further into the issue but I did find some things in other forums which didn't help either. The most common thing was a Trojan which changes the image path of the BITS service by misspelling SystemRoot like fsystemRoot but that wasn't my issue either. One thing I suspect is that Microsoft may have incorrectly downloaded the latest Windows Update Agent to my computer. If the agent is for Windows 7/8, then that could be the issue. I'm going to take a look at group policy also to see if there's a setting that turns off the service. If unsuccessful, I may send Microsoft an error report via the Windows Updates site.
I don't think I mentioned that this is actually for a Windows XP SP3 computer but looking at Services on Vista and Windows 7 looks very much the same. You see I have several computers with different OSs. By the way, I did a full scan of the computer in Safe Mode and it found nothing but two tracking cookies. So, I highly doubt it's a virus. Also comtemplating a registry clean at this point. The computer in question never had one done and it is several years old!