  Context: Tidal and rotational deformation of fluid giant extra-solar planets
may impact their transit light curves, making the $k_2$ Love number observable
in the upcoming years. Studying the sensitivity of $k_2$ to mass concentration
at depth is thus expected to provide new constraints on the internal structure
of gaseous extra-solar planets. Aims: We investigate the link between the mean
polar moment of inertia $N$ of a fluid, stably layered extra-solar planet and
its $k_2$ Love number, aiming at obtaining analytical relationships valid, at
least, for some particular ranges of the model parameters. We also seek a
general, approximate relationship useful to constrain $N$ once observations of
$k_2$ will become available. Methods: For two-layer fluid extra-solar planets,
we explore the relationship between $N$ and $k_2$ by analytical methods, for
particular values of the model parameters. We also explore approximate
relationships valid over all the possible range of two-layer models. More
complex planetary structures are investigated by the semi-analytical propagator
technique. Results: A unique relationship between $N$ and $k_2$ cannot be
established. However, our numerical experiments show that a `rule of thumb' can
be inferred, valid for complex, randomly layered stable planetary structures.
The rule robustly defines the upper limit to the values of $N$ for a given
$k_2$, and agrees with analytical results for a polytrope of index one and with
a realistic non-rotating model of the tidal equilibrium of Jupiter.
