  The measurements of the top quark flavor changing neutral current
interactions are one of the most important goals of the top quark physics
program in the present and the future collider experiments. These measurements
provide direct information on non-standard interactions of the top quark.
Within the framework of new physics beyond the Standard Model, these
interactions can be defined by an effective Lagrangian. In this study, we have
investigated the potential of the future $\mu p$ colliders on the top quark
flavor changing neutral current interactions through the subprocesses $\gamma q
\rightarrow t \rightarrow W b$ where $q=u,c$. These subprocesses have been
produced through the main reaction $\mu p \rightarrow \mu \gamma p \rightarrow
\mu W b X $ at the LHC$-\mu p$, the FCC$-\mu p$ and the SPPC-$\mu p$. For the
main reaction, the total cross sections have been calculated as a function of
the anomalous $ tq\gamma $ couplings. In addition, sensitivities on BR($t
\rightarrow q \gamma$) at $95\%$ Confidence Level have been calculated. We
obtain that the best constraints on BR($t \rightarrow q \gamma$) are at the
order of 10$^{-7}$ which is four orders of magnitude better than the LHC's
experimental results.
