We consider the constraints for a curvaton with mass m ~ 1 TeV and show that they are not consistent with a purely quadratic potential. Even if the curvaton self-interactions were very weak, they must be accounted for as they affect the dynamical evolution of the curvature perturbation. We show that the only TeV-mass curvaton interaction potential that yields the correct perturbation amplitude, decays before the dark matter freeze-out, and does not give rise to non-Gaussian perturbations that are in conflict with the present limits, is given by V_int= sigma^8/M^4. The decay width of the curvaton should be in the range Gamma= 10^-15...10^-17 GeV. The model typically predicts large non-linearity parameters f_NL and g_NL that should be observable by the Planck satellite. We also discuss various physical possibilities to obtain the required small curvaton decay rate.