We present salient results of the analysis of the geometrical structure of a large fully equilibrated ensemble of nearly jammed packings of linear freely jointed chains of tangent hard spheres generated via extensive Monte Carlo simulations. In spite the expected differences due to chain connectivity, both the pair-correlation function and the contact network for chain packings are found to strongly resemble those in packings of monomeric hard spheres at the maximally random jammed (MRJ) state. A remarkable finding of the present work is the tendency of chains to form closed loops at the MRJ state as a consequence of chain collapse. Our simulations on disordered nearly jammed chain packings yield an average coordination number of 6, which fulfills the isostaticity condition and is in excellent agreement with the corresponding simulation [A. Donev, S. Torquato, and F. H. Stillinger, Phys. Rev. E 71, 011105 (2005)] and experimental [T. Aste, M. Saadatfar, and T. J. Senden, Phys. Rev. E 71, 061302 (2005)] findings for jammed packings of monatomic hard spheres. An exact correspondence between the statistical-mechanical ensembles of monomeric spheres and of hard-sphere chains offers insights regarding the structure and topology of the contact network of hard-sphere systems at the MRJ state.