  We investigate flavour violating four Fermi Standard Model Effective Field
Theory (SMEFT) operators of dimension-six that can be probed via $tc
~(\bar{t}c+t\bar{c})$ production at the multi-TeV muon collider. We study
different FCNC and FCCC processes related to $B$, $B_s$, $K$ and $D$ decays and
mixings, sensitive to these operators and constrain the corresponding
couplings. The tensor operator turns out to be most tightly bound. We perform
event simulation of the final state signal from $tc$ production together with
the SM background to show that operators after flavour constraint can reach the
discovery limit at 10 TeV muon collider. We further adopt the optimal
observable technique (OOT) to determine the optimal statistical sensitivity of
the Wilson coefficients and compare them with the flavour constraints. We use
the limits to predict the observational sensitivities of the rare processes
like $K_L \to \pi_0 \ell \ell$, $D_0 \to \mu\mu$, $t \to c\ell\ell$.
