  In this paper we study the Lepton Flavor Violating $Z$ boson decays $Z \to
\tau \mu$ and $Z \to \tau e$ in the context of low scale seesaw models with new
heavy Majorana neutrinos whose masses could be reachable at the LHC. Our
computations of the decay rates are done in the particular realization given by
the Inverse Seesaw Model with six extra heavy neutrinos which are
quasi-degenerate in three pseudo-Dirac pairs. In particular, we focus on
scenarios that are built ad-hoc to produce suppressed rates in all the
processes involving $\mu$-$e$ transitions, given the fact that these are by far
the most strongly constrained by present data. We will fully explore the $Z \to
\tau \mu$ and $Z \to \tau e$ rates, together with a set of observables that we
find to be the most constraining ones, and we will conclude that sizable rates
of up to $2 \times 10^{-7}$, accessible at future colliders, can be reached in
this model for Majorana masses in the few TeV range, potentially reachable at
LHC.
