Over the last few years, several studies have examined the potential of using rigid-sphere microphone arrays for reconstructing sound fields with near-field acoustic holography (NAH). The existing methods provide a reconstruction of the sound field based on a spherical harmonic expansion. However, because of the basis functions used, the reconstruction can only be performed in spherical surfaces concentric to the array and inside the source-free region, which imposes a severe limitation on the applicability of the technique due to geometrical constrains. In this paper, a method based on an equivalent source model is proposed, where a combination of point sources is used to describe the incident sound field on the array. This method makes it possible to reconstruct the entire sound field at any point of the source-free domain without being restricted to a spherical surface. Additionally, this approach adds versatility (the reconstruction can be based on the microphones that are closer to the source, for a better conditioning of the problem, or also use non-uniform sampling). The method is presented, examined numerically and experimentally, and compared to the existing methods based on a spherical harmonic expansion.