  We investigate the exoplanet candidate TOI-1422b, which was discovered by the
TESS space telescope around the high proper-motion G2V star TOI-1422 ($V=10.6$
mag), 155pc away, with the primary goal of confirming its planetary nature and
characterising its properties. We monitored TOI-1422 with the HARPS-N
spectrograph for 1.5 years to precisely quantify its radial velocity variation.
The radial velocity measurements are analyzed jointly with TESS photometry and
we also check for blended companions through high-spatial resolution images
using the AstraLux instrument. We estimate that the parent star has a radius
and a mass of $R^*=1.019_{-0.013}^{+0.014} R_{\odot}$,
$M^*=0.981_{-0.065}^{+0.062} M_{\odot}$, respectively. Our analysis confirms
the planetary nature of TOI-1422b and also suggests the presence of a
Neptune-mass planet on a more distant orbit, the candidate TOI-1422c, which is
not detected in TESS light curves. The inner planet, TOI-1422b, orbits on a
period $P_{\rm b}=12.9972\pm0.0006$ days and has an equilibrium temperature
$T_{\rm eq, b}=867\pm17$ K. With a radius of $R_{\rm b}=3.96^{+0.13}_{-0.11}
R_{\oplus}$, a mass of $M_{\rm b}=9.0^{+2.3}_{-2.0} M_{\oplus}$ and,
consequently, a density of $\rho_{\rm b}=0.795^{+0.290}_{-0.235}$ g cm$^{-3}$,
it can be considered a warm Neptune-size planet. Compared to other exoplanets
of similar mass range, TOI-1422b is among the most inflated ones and we expect
this planet to have an extensive gaseous envelope that surrounds a core with a
mass fraction around $10\%-25\%$ of the total mass of the planet. The outer
non-transiting planet candidate, TOI-1422c, has an orbital period of $P_{\rm
c}=29.29^{+0.21}_{-0.20}$ days, a minimum mass, $M_{\rm c}\sin{i}$, of
$11.1^{+2.6}_{-2.3} M_{\oplus}$, an equilibrium temperature of $T_{\rm eq,
c}=661\pm13$ K and, therefore, if confirmed, it could be considered as another
warm Neptune.
