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Our main aim in this paper is to present detailed studies to probe the topquark flavor changing neutral current (FCNC) interactions at $tqg$, $tq\gamma$,$tqH$ and $tqZ (\sigma^{\mu \nu}, \gamma_{\mu})$ vertices in the triple-topsignal $p p \to t \bar t t \, (\bar t t \bar t)$ at the high energy proposal ofLarge Hadron Collider (HE-LHC) and future circular hadron-hadron collider(FCC-hh). To this end, we investigate the production of three top quarks whicharises from the FCNC couplings taken into account the fast simulation at$\sqrt{s} = 27$ TeV of HE-LHC and 100 TeV of FCC-hh considering the integratedluminosities of 10, 15 and 20 ab$^{-1}$. All the relevant backgrounds areconsidered in a cut based analysis to obtain the limits on the anomalouscouplings and the corresponding branching ratios. The obtained exclusion limitson the coupling strengths and the branching ratios are summarized and comparedin details with the results in the literature, namely the most recent directLHC experimental limits and HL-LHC projections as well. We show that, forhigher energy phase of LHC, a dedicated search for the top quark FCNC couplingscan achieve much better sensitivities to the triple-top signal than other topquark production scenarios. We found that the limits for the branching ratiosof $tqg$ and $tqH$ transitions could reach an impressive sensitivity and theobtained 95\% CL limits are at least three orders of magnitude better than thecurrent LHC experimental results as well as the existing projections of HL-LHC.

Probing top quark FCNC couplings in the triple-top signal at the high energy LHC and future circular collider