Abstract: We consider the effect of nonstandard neutrino interactions (NSI, for short) on the propagation of neutrinos through the supernova (SN) envelope within a three-neutrino framework and taking into account the presence of a neutrino background. We find that for given NSI parameters, with strength generically denoted by epsilon(ij), neutrino evolution exhibits a significant time dependence. For vertical bar epsilon(tau tau)vertical bar greater than or similar to 10(-3) the neutrino survival probability may become sensitive to the V-23 octant and the sign of epsilon(tau tau). In particular, if epsilon(tau tau) greater than or similar to 10(-2) an internal I-resonance may arise independently of the matter density. For typical values found in SN simulations this takes place in the same dense-neutrino region above the neutrinosphere where collective effects occur, in particular, during the synchronization regime. This resonance may lead to an exchange of the neutrino fluxes entering the bipolar regime. The main consequences are (i) bipolar conversion taking place for normal neutrino mass hierarchy and (ii) a transformation of the flux of low-energy v(e), instead of the usual spectral swap.