  Primordial black holes could have been formed in the early universe from
sufficiently large cosmological perturbations re-entering the horizon when the
Universe is still radiation dominated. These originate from the spectrum of
curvature perturbations generated during inflation at small-scales. Because of
the non-linear relation between the curvature perturbation $\zeta$ and the
overdensity $\delta\rho$, the formation of the primordial black holes is
affected by intrinsic non-Gaussianity even though the curvature perturbation is
Gaussian. We investigate the impact of this non-Gaussianity on the critical
threshold $\delta_c$ which measures the excess of mass of the perturbation,
finding a relative change with respect to the value obtained using a linear
relation between $\zeta$ and $\delta\rho$, of a few percent. This shows that
the value of the critical threshold is rather robust against non-linearities.
The same holds also when cosmologically interesting values of local primordial
non-Gaussianity are added to the curvature perturbation.
