Multi-piece golf balls having a solid core made of a foamed composition and a cover are provided. Preferably, the core is dual-layered having has a foam inner core (center) and surrounding thermoset or thermoplastic outer core layer. Preferably, a polyurethane foam composition containing mineral filler particulate, for example, nanoclay particles, is used to form the foam center. The concentration of clay particulate is preferably in a range of 0.1 to 60% by weight. The surrounding outer core layer may be made from non-foamed or foamed compositions. For example, polybutadiene rubber or highly neutralized olefin acid copolymers may be used in the outer core layer. The core layers have different hardness gradients and specific gravity values.