  Data from transit light curves, radial velocity and transit timing
observations can be used to probe the interiors of exoplanets beyond the mean
density, by measuring the Love numbers $h_2$ and $k_2$. The first indirect
estimate of $k_2$ for an exoplanet from radial velocity and transit timing
variations observations has been performed by taking advantage of the
years-spanning baseline. Not a single measurement of $h_2$ has been achieved
from transit light curves, mostly because the photometric precision of current
observing facilities is still too low. We show that the Imaging Spectrograph
instrument on-board the Hubble Space Telescope could measure $h_2$ of the hot
Jupiter WASP-121b if only few more observations were gathered. We show that a
careful treatment of the noise and stellar limb darkening must be carried out
to achieve a measurement of $h_2$. In particular, we find that the impact of
the noise modelling on the estimation of $h_2$ is stronger than the impact of
the limb darkening modelling. In addition, we emphasize that the wavelet method
for correlated noise analysis can mask limb brightening. Finally, using
presently available data, we briefly discuss the tentative measurement of $h_2
= 1.39^{+0.71}_{-0.81}$ in terms of interior structure. Additional observations
would further constrain the interior of WASP-121b and possibly provide insights
on the physics of inflation. The possibility of using the approach presented
here with the Hubble Space Telescope provides a bridge before the high-quality
data to be returned by the James Webb Space Telescope and PLATO telescope in
the coming decade.
