We study the scale dependence of the non-linearity parameters f_NL and g_NL in curvaton models with self-interactions. We show that the spectral indices n_{f_NL} = d ln|f_NL|/d ln k and n_{g_NL} = d ln|g_NL|/d ln k can take values much greater than the slow-roll parameters and the spectral index of the power spectrum. This means that the scale-dependence of the bi and trispectrum could be easily observable in this scenario with Planck, which would lead to tight additional constraints on the model. Inspite of the highly non-trivial behaviour of f_NL and g_NL in the curvaton models with self-interactions, we find that the model can be falsified if g_NL(k) is also observed.