Using density-functional theory calculations, we studied the electronic and magnetic properties of zigzag graphene nanoribbons (ZGNRs) with gold (Au) atoms embedded into different sites of the ZGNRs. Strong site dependence was found, and the system had the ferromagnetic or antiferromagnetic ground state depending on the Au atom position. Spin gapless semiconductor (SGS) behavior was observed when the Au atom was embedded into the center and edge sites of the ZGNRs. The simulations showed that the electronic structure of the ribbon strongly depends on ZGNR width, but the SGS behavior is always present when the Au atom is embedded into the center and edge sites. The SGS properties were also found to be dependent on impurity atom concentration, so that they can be tuned by either selecting the proper positions of Au atoms or changing their concentration. Our results suggest a flexible way of designing SGSs, which could be used in various spintronic, electronic, and optoelectronic applications.