  Lepton flavor violating Higgs boson decays (LFVHD) are studied in the context
of the Minimal Supersymmetric Standard Model (MSSM) being embeded in a mSUGRA
scenario that is 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 mSUGRA 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.
