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 ζ and the overdensity δρ, 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 δc which measures the excess of mass of the perturbation, finding a relative change with respect to the value obtained using a linear relation between ζ and δρ, of a few percent. This shows that the value of the critical threshold is rather robust against non-linearities. This allows a computation of the abundance of primordial black holes which is more precise than using the critical amplitude of the peak, since the latter is more sensitive to the local feature of the shape of the perturbation. The same holds also when cosmologically interesting values of local primordial non-Gaussianity are added to the curvature perturbation..