Multi-layered golf balls containing a dual-core structure are provided. The core structure includes an inner core (center) comprising a foam composition, preferably foamed polyurethane. The outer core layer is preferably formed from a non-foamed thermoset composition such as polybutadiene rubber. The core layers have different hardness and specific gravity levels. The specific gravity (density) of the foam inner core is preferably less than the density of the outer core layer. The core assembly preferably has a positive hardness gradient extending across the entire assembly. The core structure and resulting ball have relatively good resiliency.