Here's what I'd do:
First with a cold engine, hood open and engine cover off, (before stating 1st check of fan clutch) I start engine. Trying to detriment general area of sound i.e front side(s), top or rear of engine. Which most common is front. So I'd take a $5 Harbor Fright stethoscope and check for any chirps, oscillating, sandy or grinding sounds. Being care not to get loose clothes, finger or tool caught in any moving parts. Listen to: Idler pulley (on blot head), tensioner pulley (on bolt head) and fan bracket bearings (on body/case/housing behind pulley/belt). I'd also check the alternator (on body/case/housing, don't short if contacts exposed), vane pump (on body/case/housing) and AC compressor (on body/case/housing). Look also to see if fan bracket pulley jumps or wobbles as in spins, in conjuction with sound.
I'd check fan clutch for proper operation before starting engine and as engine warms up, while I'm listening with stethoscope.
Next I'd remove fan belt do more checks. While removing belt, by way of tension 14mm left hand bolt. Note pull of tensioner, it should pull against you strong (about 12lb)
Then while fan belt off. Start engine. Is sound still present at front of engine. Then I'd suspect it's from Timing belt or it's pulleys or water pump.
Also worth a note, a wonky of partially stuck PCV valve of bad vacuum lines/hoses. Can make stranger sounds. So do inspect them.
Also inspect air filter, make sure in good condition and that it's rubber seal is good (not curled). Also air pipe to throttle body connected properly, making perfect seal 360 degrees (underside I find curled, all to often).
Also look very close at any work done, just before sounds started.
If all checks-out, as good. Hose fan belt with water, while engine running. Does it squeal as water drys. Replace the fan belt with OEM.