Sort of sounds like the fan clutch, but also sounds a bit different somehow than what I’ve heard before. Can’t quite put my finger on it. Also, fan clutch should not care if the engine is cold or hot.
The fan has an electrical connection on it - you could disconnect it then run it for a few minutes and see if the noise stops. The fan clutch noise that you commonly hear is a solenoid that is trying (and failing) to engage, so it just keeps hammering away like a ticking clock. If you disconnect it, there will be no voltage to the fan to make the noise. If that stops the noice, the fan clutch needs to be replaced.
If doesn’t stop, and it sounds like it is coming from that area, turn it off and try rotating the fan by hand and see if it is maybe just ticking part of the shroud (which might be causing something else to rattle). When the engine mounts start to go, the motor can shift enough that the fan starts to hit the shroud. Again - hot/cold engine should not make any difference.
Also - check for codes. If you don’t have a code reader, any parts store will be able to at least give you a generic code read for free.