The only two devices in the system with moving parts are the fan or the hard drive.

It's either the drive spinning up, or the fan spinning up.  

Check the fans for dust buildup -- just a little can cause them to be less than balanced, and rumble.

No, not necessarily.  The drive does spin at somewhere between 5400 and 7200 rpm -- it will vibrate to some extent.  If it's objectionable, replace it with a solid state drive, which will be silent in operation.

I have found that the start up rumble is from the hard drive.  Does this indicate impending failure of the hard drive?

The rumble is not objectionable.  I am concerned because it just started last week.  Is there a diagnostic tool I can use to make sure the drive is not going to fail?

Yes.  F12 at powerup will get you into the Dell diagnostics -- run an extended test on the hard drive after the quick tests run, and always keep current backups -- there are two types of drives:  those that have failed, and those that will.

The quick test found two errors,  It did not identify the errors, nor offer an option to repair them.  After the Quick Test an extended diagnostic test was not offered.  I did not see any menus or buttons that would allow me to navigate beyond the initial diagnostic screen.

The Dell diagnostics (NOT the support assist) will always return numeric errors - 2000-0142, etc.  Is that what you ran?

These run outside of Windows - not under Windows.

If it had been a hard drive error, you would have seen something like 2000-0142.  If you didn't, it sounds like the drive passed its test.

If the noise worsens, and you're in warranty, call Dell;  out of warranty, replace the  hard drive with a 2.5" SATA SSD, which will run silently.