  Lepton flavor violating Higgs boson decays (LFVHD) are studied in the context
of the Minimal Supersymmetric Standard Model (MSSM) enlarged with three right
handed neutrinos and their supersymmetric partners, and with the neutrino
masses being generated by the seesaw mechanism. We compute the partial widths
for these decays to one-loop order and analyze numerically the corresponding
branching ratios in terms of the MSSM and seesaw parameters. We analyze in
parallel the lepton flavor changing $\l_j\to l_i \gamma$ decays and explore the
maximum predicted rates for LFVHD, mainly for $H^0,A^0 \to \tau {\bar \mu}$
decays, by requiring compatibility with neutrino and $BR(\l_j\to l_i \gamma)$
data. We find LFVHD ratios of up to $10^{-5}$ in some regions of the
MSSM-seesaw parameter space.
