Observational extragalactic catalogs over wide sky areas are essential for uncovering the large-scale structure of the Universe. They allow, among other things, cosmological studies and density analyses that impose strong constraints on models of galaxy formation and evolution. By taking advantage of the wide field images and the 12 optical bands of the Southern Photometric Local Universe Survey (S-PLUS), we aim to provide a catalog of galaxies located, in projection, toward the Fornax galaxy cluster, within \(\sim\) 5 virial radii in right ascension (R.A.) and \(\sim\) 3 virial radii in declination (Dec) around NGC,1399, the dominant galaxy of the cluster. Such a catalog will allow unprecedented large-scale structure studies in that sky region. We developed supervised deep-learning algorithms, utilizing neural networks complemented by dimensionality-reduction techniques, to classify and separate spurious objects, stars and galaxies in a photometric catalog previously built for the S-PLUS Fornax Project (S+FP). That catalog was built using a combination of SExtractor configurations optimized for galaxy detection and characterization. A catalog of 119,580 galaxies was obtained in the direction of the Fornax cluster containing photometric information in the 12 optical bands of S-PLUS complemented with GALEX (UV), VHS-VISTA (NIR), and AllWISE (MIR) data. We estimate photometric redshifts (σ\(_{NMAD}\) \(\sim\) 0.0219) with a lower limit of z\(_{lim}\) \(\sim\) 0.03. Stellar masses, star formation rates (SFRs), and D4000\(_N\) index estimates were obtained through a machine-learning approach, by matching S-PLUS photometric data to SDSS spectroscopic data. The completeness of the catalog (72%) was calculated by comparing it with mock catalogs. Taking into account our z\(_{lim}\), we were able to identify 119,230 background galaxies and to find 350 candidates to be Fornax members or infalling galaxies, which were not included in our compilation of 1,005 galaxies previously reported in the literature. We were also able to classify the galaxies in our catalog as quiescent (43%), star forming (39%), and transition (18%) galaxies. In addition, 181 emission line galaxy (ELG) candidates were identified using the filter J0660. The spatial distribution of the galaxies in our catalog shows projected overdensities that match 158 background clusters identified by eROSITA. This confirms the robustness of our catalog in tracing real structures. In that context, we expect the extragalactic catalog of the S+FP to allow us to better understand the large-scale structure in the direction of the Fornax cluster and to identify the substructures that are feeding Fornax.