  In this paper, we consider a single top quark production in association with
a neutral gauge boson (Z boson or Photon) at the future electron-positron
colliders. We utilize these channels to probe the top quark Flavour Changing
Neutral Currents (FCNC) interactions in $tqZ$ and $tq\gamma$ vertices. We
perform two separate analyses for top-Z-jet and top-$\gamma$-jet channels. The
Standard Model Effective Field Theory (SMEFT) approach is employed to search
for these anomalous couplings. We consider parton showering, hadronization and
fast detector simulation in our study and use a cut-and-count technique to
separate the signal from the Standard Model (SM) background processes. The
upper limits on the FCNC couplings at $95\%$ confidence level for the different
integrated luminosities are obtained. It is shown that the future lepton
collider would be able to probe the FCNC branching fractions to
$\rm{Br}(t\rightarrow q \gamma)< 10^{-4}$ and $\rm{Br}(t\rightarrow q Z) <
10^{-4}$ with $3$ $\rm{ab}^{-1}$of integrated luminosity of data at a center of
mass energy of $350$ GeV.
