  In this study, we investigate the imprints of $C\!P$ violation in certain
$\eta$ muonic decays that could arise within the Standard Model effective field
theory. In particular, we study the sensitivities that could be reached at
REDTOP, a proposed $\eta$ facility. After estimating the bounds that the
neutron EDM places, we find still viable to discover signals of $C\!P$
violation via the muons' polarization in $\eta\to\mu^+\mu^-$ decays at REDTOP,
with a single effective operator as its plausible source.
