Reads like the fan clutch is failing.
CEL(check engine light) wouldn't be related to that since fan clutch is a mechanical part.
But engine could be overheating because of that and that can cause temp gauge to go up and CEL to come on, but CEL could be unrelated so get the code ASAP, auto parts stores read codes for free, WRITE THEM ALL DOWN, but don't buy anything until you post the codes here, codes don't always mean what people think.
Start engine cold and let it run for 30-60 seconds, then shut it off.
Open hood and try to spin the fan, it should spin easily, if not fan clutch is bad, replace it.
Normally when you start a cold engine with mechanical fan you will hear the fan noise, after about 10 to 20 seconds the noise should diminish so you don't hear it.
Inside the fan clutch is a fluid and valve, when cold the fluid is thick, when you start the engine the fan clutch start turning and the fan blade grabs alot of air, the noise you hear, the fluid warms up quickly, and the valve opens, this loosens the clutches grip on the fan blade so it starts spinning slower and doesn't grab as much air so quiets down after about 20 seconds.
On the front of the fan clutch is a bi-metal spring, this spring holds the internal valve open allowing fan blade to spin at slower speed than fan belt/water pump, so quietly.
As radiator heats up it heats up the bi-metal spring, when heated the spring expands and slowly closes the internal valve causing the clutch to engage the fan blade more and more.
If you shut off engine after radiator is warned up the fan blade should not spin easily, it will move when pushed or pulled but not spin.
You could also have an alternator or other pulley bearing starting to fail, these can make similar noise, remove fan belt and spin each pulley to see if it could be cause of the noise.