  The search for Flavons with a mass of $\mathcal{O}$(1) TeV at current and
future colliders might probe low-scale flavor models. We are interested in the
simplest model that invokes the Froggatt-Nielsen (FN) mechanism with an Abelian
flavor symmetry, which includes a Higgs doublet and a FN complex singlet.
Assuming a CP conserving scalar potential, there are a $CP$-even $H_F$ and a
$CP$-odd $A_F$ Flavons with lepton flavor violating (LFV) couplings. The former
can mix with the standard-model-like Higgs boson, thereby inducing tree-level
LFV Higgs interactions that may be at the reach of the LHC. We study the
constraints on the parameter space of the model from low-energy LFV processes,
which are then used to evaluate the Flavon decay widths and the $gg\to \phi \to
\tau\mu$ ($\phi=H_F,\,A_F$) production cross section at hadron colliders. After
imposing several kinematic cuts to reduce the SM main background, we find that
for $m_{H_F}$ about 200-350 GeV, the decay $H_F \to \tau \mu$ might be at the
reach of the LHC for a luminosity in the range 1-3 ab$^{-1}$, however, a
luminosity of the order of 10 ab$^{-1}$ would be required to detect the $A_F
\to \tau \mu$ decay. On the other hand a future 100 TeV $pp$ collider could
probe masses as high as $\mathcal{O}$(10) TeV if it reaches an integrated
luminosity of at least 20 ab$^{-1}$. Therefore, the 100 TeV Collider could work
as a Flavon factory.
