  We analyze the most natural formulations of the minimal lepton flavour
violation hypothesis compatible with a type-I seesaw structure with three heavy
singlet neutrinos N, and satisfying the requirement of being predictive, in the
sense that all LFV effects can be expressed in terms of low energy observables.
We find a new interesting realization based on the flavour group $SU(3)_e\times
SU(3)_{\ell+N}$ (being $e$ and $\ell$ respectively the SU(2) singlet and
doublet leptons). An intriguing feature of this realization is that, in the
normal hierarchy scenario for neutrino masses, it allows for sizeable
enhancements of $\mu \to e$ transitions with respect to LFV processes involving
the $\tau$ lepton. We also discuss how the symmetries of the type-I seesaw
allow for a strong suppression of the N mass scale with respect to the scale of
lepton number breaking, without implying a similar suppression for possible
mechanisms of N production
