In case the water pump is at the end of it's lifetime, you will hear the water pump bearing, and also coolant will come out at the front of the water pump behind the pulley. I changed the water pump on my E36 M3. I could hear the bearing noise when engine was running, no coolant leaks yet, and when I had it out of the engine and tried to turn the pulley by hand, it was almost impossible, bearing was almost seized.

Water pump failure signs by GATES-HOW TO DETECT A BAD WATER PUMP?

The following signs indicate that a water pump needs to be replaced:
1. Weep hole leakage
Appearance: The pump's internal mechanical seal seals the shaft towards the cooling circuit, protecting the bearings by preventing coolant from passing into the bearing assembly. When a water pump is new, some seepage from the weep hole is normal as it takes about ten minutes of operation for the mechanical seal to properly seat itself (break-in period). More pronounced seepage and drips from the weep hole after this break-in period or a large coolant bleed mark around the weep hole are abnormal and indicate impending water pump failure.

Cause: Contaminated coolant is the main cause of weep hole leakage.

Solution: Thoroughly flush the cooling system before installing the new pump and refill the system with the correct vehicle manufacturer’s recommended coolant.

Never dry run a water pump, not even for a few seconds. Dry running of the pump ruins its mechanical seal.

2. Leakage from the mounting surface
Appearance: Seepage, drips or large coolant bleed marks on or around the mounting surface or on the housing.
Cause: Improper water pump installation or improper use of seals/gaskets or sealant.
Solution: In case of a recently installed new water pump: carefully remove, check and re-install the water pump. Strictly follow the torque specifications. Make sure seals/gaskets are in perfect condition and are installed correctly. When sealant is prescribed, clean the rims of the part and the mounting surface and apply new sealant evenly along the edge of the part. If the leakage is not the consequence of an improperly installed new water pump, the pump must be replaced immediately.

3. Rust and corrosion
Appearance: Rust and corrosion on the surfaces of the water pump, e.g. corrosion of the impeller fins makes the pump ineffective in moving the coolant.
Cause: Contaminated coolant, non-compatible coolant or mixing coolants of different chemistries. Another possible cause may be a defective pressure cap causing air bubbles, accelerating the rust process.
Solution: Replace the water pump. Thoroughly flush the cooling system before installing the new pump and refill the system with the correct vehicle manufacturer’s recommended coolant. Inspect the pressure cap and replace if defective.

4. Deposit build-up
Appearance: Deposits, sludge and scale build up on the inside, clogging the water pump and hampering proper functioning of its individual components.
Cause: Contaminated coolant, non-compatible coolant or mixing coolants of different chemistries.
Solution: Replace the water pump. Thoroughly flush the cooling system before installing the new pump and refill the system with the correct vehicle manufacturer’s recommended coolant.

5. Cavitation
Appearance: Vapour cavities (“bubbles”) in the coolant collapse with explosive force, pockmarking the pump’s individual components. The pockmarked areas then corrode away.
Cause: The formation of bubbles at the coolant inlet of the pump. As the pressure increases in the system, the bubbles implode and cause harm.
Solution: Replace the water pump. Thoroughly flush the cooling system before installing the new pump and refill the system with the correct vehicle manufacturer’s recommended coolant.

6. Damaged bearing
Appearance: With the engine off, check the condition of the bearing by looking for any side-to-side play at the shaft. Apply hand pressure – there should be no play at all. Also, rumbling or screeching noises coming from the water pump indicate a worn bearing.
Cause: A misaligned belt causes excessive wear. An over-tensioned belt causes overload on the bearing, accelerating wear. A damaged mechanical seal allows coolant to leak through the bearings and wash away the lubricator.
Solution: Replace the water pump and make sure to inspect the belt drive system: belt, tensioner, pulleys, belt tension and alignment.
Never compromise on water pump quality! Cheap water pumps do not follow the OE prescription in terms of mechanical seal and bearing quality.

7. Damaged or broken shaft
Appearance: The shaft is bent or broken. A clean break indicates an “instantaneous” fracture caused by sudden overload or imbalance. If the broken shaft is discoloured (usually blue), the damage was gradual, indicating excessive heat build-up before the shaft broke.
Cause: A misaligned belt causes excessive wear. An over-tensioned belt imposes a powerful bending force on the shaft resulting in shaft bending or early shaft fracture. Excessive vibrations due to lack of maintenance.
Solution: Replace the water pump and make sure to inspect the belt drive system: belt, tensioner, pulleys, belt tension and alignment. If equipped with a water pump mounted fan, inspect the fan/fan clutch assembly. The fan may not be squarely mounted on the shaft. A bent or damaged fan, a worn spacer, or a worn or damaged fan clutch could also be factors contributing to the break.

with pics here from Gates https://www.gatestechzone.com/en/pro...-failure-signs