  We study the flavor-changing decay $h\to \tau \mu$ with $\tau=\tau^-+\tau^+$
and $\mu=\mu^-+\mu^+$ of a Higgs boson at future hadron colliders, namely: a)
High Luminosity Large Hadron Collider, b) High Energy Large Hadron Collider and
c) Future hadron-hadron Circular Collider. The theoretical framework adopted is
the Two-Higgs-Doublet Model type III. The free model parameters involved in the
calculation are constrained through Higgs boson data, Lepton Flavor Violating
processes and the muon anomalous magnetic dipole moment; later they are used to
analyze the branching ratio of the decay $h\to\tau\mu$ and to evaluate the
$gg\to h$ production cross section. We find that at the Large Hadron Collider
is not possible to claim for evidence of the decay $h\to\tau\mu$ achieving a
signal significance about of $1.46\sigma$ by considering its final integrated
luminosity, 300 fb$^{-1}$. More promising results arise at the High Luminosity
Large Hadron Collider in which a prediction of 4.6$\sigma$ when an integrated
luminosity of 3 ab$^{-1}$ and $\tan\beta=8$ are achieved. Meanwhile, at the
High Energy Large Hadron Collider (Future hadron-hadron Circular Collider) a
potential discovery could be claimed with a signal significance around
$5.04\sigma$ ($5.43\sigma$) for an integrated luminosity of 3 ab$^{-1}$ and
$\tan\beta=8$ (5 ab$^{-1}$ and $\tan\beta=4$).
