  In this work, we adopt the Nambu-Jona-Lasinio (NJL) model that ensures the
nuclear matter saturation properties to study the density dependence of the
symmetry energy. With the interactions constrained by the chiral symmetry, the
symmetry energy shows novel characters different from those in conventional
mean-field models. First, the negative symmetry energy at high densities that
is absent in relativistic mean-field (RMF) models can be obtained in the RMF
approximation by introducing a chiral isovector-vector interaction, although it
would be ruled out by the neutron star (NS) stability. Second, with the
inclusion of the isovector-scalar interaction the symmetry energy exhibits a
general softening at high densities even for the large slope parameter of the
symmetry energy. The NS properties obtained in the present NJL model can be in
accord with the observations. The NS maximum mass obtained with various
isovector-scalar couplings and momentum cutoffs is well above the $2M_\odot$,
and the NS radius obtained well meets the limits extracted from recent
measurements. In particular, the significant reduction of the canonical NS
radius occurs with the moderate decrease of the slope of the symmetry energy.
