  The near-Earth objects with low-Jovian Tisserand invariant ($T_J$) represent
about 9 per cent of the known objects orbiting in the near-Earth space, being
subject of numerous planetary encounters and large temperature variations. We
aim to make a spectral characterization for a large sample of NEOs with $T_J$
$\leq$ 3.1. Consequently, we can estimate the fraction of bodies with a
cometary origin. We report new spectral observations for 26 low-T$_J$ NEOs. The
additional spectra, retrieved from different public databases, allowed us to
perform the analysis over a catalogue of 150 objects. We classified them with
respect to Bus-DeMeo taxonomic system. The results are discussed regarding
their orbital parameters. The taxonomic distribution of low-$T_J$ NEOs differs
from the entire NEOs population. Consequently, $T_J$$\sim$3 can act as a
composition border too. We found that 56.2 per cent of low-T$_J$ NEOs have
comet-like spectra and they become abundant (79.7 per cent) for T$_J$ $\leq$
2.8. 16 D-type objects have been identified in this population, distributed on
orbits with an average T$_J$ = 2.65 $\pm$ 0.6. Using two dynamical criteria,
together with the comet-like spectral classification as an identification
method and by applying an observational bias correction, we estimate that the
fraction of NEOs with a cometary nature and H $\in$ (14, 21) mag has the lower
and upper bounds (1.5 $\pm$ 0.15) and (10.4 $\pm$ 2.2) per cent. Additionally,
our observations show that all extreme cases of low-perihelion asteroids (q
$\leq$ 0.3 au) belong to S-complex.
