  Lyman-$\alpha$ nebulae are usually found in massive environments at high
redshift ($z > 2$). The origin of their Lyman-$\alpha$ (Lya) emission remains
debated. Recent polarimetric observations showed that at least some Lya sources
are polarized. This is often interpreted as a proof that the photons are
centrally produced, and opposed to the scenario in which the Lya emission is
the cooling radiation emitted by gas heated during the accretion onto the halo.
We suggest that this scenario is not incompatible with the polarimetric
observations. In order to test this idea, we post-process a radiative
hydrodynamics simulation of a blob with the MCLya Monte Carlo transfer code. We
compute radial profiles for the surface brightness and the degree of
polarization and compare them to existing observations. We find that both are
consistent with a significant contribution of the extragalactic gas to the Lya
emission. Most of the photons are centrally emitted and scattered inside the
filament afterwards, producing the observed high level of polarization. We
argue that the contribution of the extragalactic gas to the Lya emission does
not prevent polarization to arise. On the contrary, we find that pure galactic
emission causes the polarization profile to be too steep to be consistent with
observations.
