What you describe is typical of a FEAD bearing that's not properly lubricated.
(front end accesory drive)
More than likely it's the tensioner arm pulley. (pass side of a 4.0L)
Background.. when a bearing is cold and in-adequatly lubed. It operates dry. And for a short time the rolling elements (***** and rollers) are loose in thier raceways. As the metal heats up it expands. As they expand & heat up, the bearings grease starts to fully coat everything as well as the clearances getting tighter. So.. the noise gets softer and goes away. Sometimes the noise will go away instantly as well.
If this is the real problem it will only get worse. Sooner or later the bearing will develope a rough sound and eventually fail chucking the belt off.
(FEAD engineer for 6 yrs