It has been speculated for a long time that neutrinos from a supernova (SN) may undergo fast flavor conversions near the collapsed stellar core. We perform a detailed study of this intriguing possibility, for the first time analyzing two time-dependent state-of-the-art three-dimensional (3D) SN models of 9 M_☉ and 20 M_☉ from recent papers of Glas et al. Both models were computed with multidimensional three-flavor neutrino transport based on a two-moment solver, and both exhibit the presence of the so-called lepton-number emission self-sustained asymmetry (LESA). The transport solution does not provide the angular distributions of the flavor-dependent neutrino fluxes, which are crucial to track the fast flavor instability. To overcome this limitation, we use a recently proposed approach based on the angular moments of the energy-integrated electron lepton-number distribution up to second order, i.e., angle-energy integrals of the difference between ν_e and ν_̄e phase-space distributions multiplied by corresponding powers of the unit vector of the neutrino velocity. With this method we find the possibility of fast neutrino flavor instability at radii smaller than ∼20 km , which is well interior to the neutrinosphere where neutrinos are still in the diffusive and near-equilibrium regime. Our results confirm recent observations in a two-dimensional (2D) (axisymmetric) SN model and in 2D and 3D models with a fixed matter background, which were computed with Boltzmann neutrino transport. However, the flavor unstable locations are not isolated points as discussed previously, but thin skins surrounding volumes where ν_̄e are more abundant than ν_e. These volumes grow with time and appear first in the convective layer of the proto-neutron star (PNS), where a decreasing electron fraction and high temperatures favor the occurrence of regions with negative neutrino chemical potential. Since the electron fraction remains higher in the LESA dipole direction, where convective lepton-number transport out from the nonconvective PNS core slows down the deleptonization, flavor unstable conditions become more widespread in the opposite hemisphere. This interesting phenomenon deserves further investigation, since its impact on SN modeling and possible consequences for SN dynamics and neutrino observations are presently unclear.