  We estimate the smallest branching ratio for the Higgs decay channel $h
\rightarrow \mu \tau$, which can be probed at an $e^+e^-$ collider and compare
it with the projected reach at the high-luminosity run of the LHC. Using a
model-independent approach, Higgs production is considered in two separate
cases. In the first case, $hWW$ and $hZZ$ couplings are allowed to be scaled by
a factor allowed by the latest experimental limits on $hWW$ and $hZZ$
couplings. In the second case, we have introduced higher-dimensional effective
operators for these interaction vertices. Keeping BR($h\to\mu\tau$) as a purely
phenomenological quantity, we find that this branching ratio can be probed down
to $\approx 2.69\times 10^{-3}$ and $\approx 5.83\times 10^{-4}$ respectively,
at the 250 GeV and 1000 GeV run of an $e^+e^-$ collider.
