It depends. The engines are the responsibility of the engine manufacturer, but the actual cowling (nacelle) can either be done by the airframer, outsourced, or sourced from the engine manufacturer.
On the 777, Boeing did most of the nacelle in-house for all three engine types. On the 787, nearly all the nacelle was outsourced (which didn't go very well). On the 747-8, the nacelle was something of a joint venture between Boeing and GE (Middle River).
But the specific mater in hand - the PW4000/112" engine as installed on the 777 - Boeing did the nacelle.