My manual transmission has been making a weird noise. When the vehicle is idling, the clutch is not engaged and it is in neutral, there is a sound coming out of the transmission case/or bell housing

it might be on the steering column

Bearing noises in a clutch system are easily distinguished by running a few simple tests. Start the engine and leave the transmission in neutral. If a transmission makes noise when it is in neutral it is often a worn input shaft bearing. Now depress the pedal until the clutch just engages. This is the point where the release bearing is just contacting the pressure plate fingers. If you begin to hear a squealing sound this is likely a worn release bearing. If a whirling noise is present when the clutch pedal is fully depressed it is probably the pilot bearing.

You need to locate the source of noise first , it seems to noise of drive line only. Which consist of flywheel , clutch system and gear box.
1) In case you are having same problem with clutch engaged or disengaged position ( while throttle is on) , then there is a possible problem with the neutral gear ( if it is not abnormal but just high in magnitude than it was previously than consider replacing the transmission oil) otherwise get the gear box checked with a transmission specialist.
2) In case you are having the noise problem only when clutch is engaged than u may consider to refit the clutch plate and pressure plate , there could be a problem of torquing.

Throw out bearing is shot

That is the noise

Never heard that before on any vehicle?

Shifting to NEUTRAL
To shift the transfer case to NEUTRAL, first make sure
the vehicle is parked so that it will not roll:
1. Set the parking brake.
2. Start the vehicle.
3. Connect the vehicle to the towing vehicle.
4. Put the transmission in NEUTRAL (N) or have
the clutch pedal pressed for vehicles with
manual transmission.
5. Shift the transfer case to 2HI.
6. Simultaneously press and hold the 2HI and
4LO buttons for 10 seconds. The red NEUTRAL
light will come on when the transfer case shift to
NEUTRAL is complete.
7. Shift the transmission to REVERSE (R) for one
second, then shift the transmission to DRIVE (D)
for one second or FIRST (1) for vehicles with
manual transmission.
8. Turn the ignition to OFF.
9. Place the transmission shift lever in PARK (P) or
FIRST (1) for vehicles with manual transmission.
10. Release the parking brake prior to towing.

************************

Shifting Out of NEUTRAL
To shift out of NEUTRAL:
1. Set the parking brake and apply the regular
brake pedal.
2. Start the vehicle with the transmission in PARK (P)
or FIRST (1) for vehicles with manual transmission.
3. Press the button for the desired transfer case position
(2HI, 4HI, AUTO 4WD or 4LO).
4. Put the transmission in NEUTRAL (N) or press the
clutch pedal for vehicles with manual transmission.
5. Shift the transmission lever to the desired position.
After the transfer case has shifted out of NEUTRAL,
the red light will go out.
6. A re-engagement sound is normal when shifting out
of NEUTRAL.

Sounds like a bad torque converter. When the torque converter is engaged, it's not "whirring" it's on decel or freewheel that it "whirs".Did you check fluid level and color?

does the clutch engage still? or no?

internal slave cylinders are known to leak , unfortunate it is behind clutch assembly , i would change both , clutch master and slave cylinder

If your certain the noise is from the transmission, you need to take this vehicle to a shop you trust. It sounds as though the fluid pump in the transmission is either starving for fluid, due to a blocked strainer, or the pump is suffering a bearing failure. If you continue to run this vehicle, you stand the chance of stranding yourself, or burning the clutches from a lack of fluid. Good Luck.

When the car is sitting idling,and the clutch is not depressed,it makes a grinding noise because the input shaft of the transmission is being forced to turn by the clutch,which is engaged,and even though the trans is in neutral,the input shaft is still turning,and this is called "neutral roll over" noise,and some of the assemblies inside the trans are freewheeling around.The noise goes away when the clutch pedal is being depressed down,because the input shaft of the transmission is no longer turning.So it does sound like an internal transmission problem.It is still possible for the throwout bearing to be bad,and it can be checked when the trans is removed from the vehicle,Remove the throwout bearing from the trans input sleeve,and press down on the bearing flange and turn it at the same time.It should feel smooth and free.If it catches or feels gravelly,then the bearing is bad and that could be your noise.Also,turn the input shaft of the trans and same thing,if it's smooth and free or if gravelly feeling......Good luck.