If you don't understand how the clutch system works, then I am just wasting my time... but here goes...
In the end of the crank, there is a pilot bearing. The tip of the transmission input shaft rests in this bearing.
On the end of the crank, is your flywheel. Bolted to the flywheel, is the clutch pressure plate.
Pressed in between the flywheel and pressure plate, is the clutch disc.
Behind this whole assembly, is the Throw Out Bearing (TOB)
The input shaft of the transmission slides through the TOB and through the center of the clutch disc, and into the pilot bearing (see above).
When at idle, and in neutral, the crank turns, which spins the flywheel/clutch disc/flexplate assembly, and spins the input shaft on the transmission.
the TOB does not spin. It has a bearing on the end that rests against the back of the pressure plate. These will sometimes squeak if they do not stay in constant contact with the Pressure Plate. If you put pressure on the clutch pedal, the actual bearing on the TOB will spin with the pressure plate, stopping the squeak.
If the noise continues, the TOB is likely NOT the cause, but if it was, you would feel the noise harmonic through the clutch pedal.
The reason I say that your trans is the culprit is because pressing the clutch pedal all the way to the floor will release the clutch disc, and it will stop spinning. Along with the input shaft of the transmission.
The two parts inside your transmission that would be causing this noise are the input shaft nose or tail bearing, or cluster shaft bearings.