We study scalar Non-Standard Neutrino Interactions (NSI) induced by moduli or flavon exchange between electrons and neutrinos. In a region with non-vanishing electron number density, they are known to determine a shift of the neutrino mass matrix. We review and extend the relevant formalism, and we update the existing limits on electron and neutrino scalar couplings. We explore the observability of scalar NSI in models of lepton masses based on flavour symmetries. We analyze models where the scalar couplings are constrained either by abelian symmetries or by modular invariance. We highlight regions of the parameter space where observable effects can occur.