The determination of the electromagnetic dipole moments of the top quark is one of the most important goals of the top quark physics program in the collider experiments. For this reason, the top quark pair production to investigate the sensitivity on the electric and magnetic dipole moments of the top quark via the process $pp\to p\gamma p\to pt\overline{t}X$ at the Large Hadron Collider (LHC), the High-Luminosity Large Hadron Collider (HL-LHC) and High-Energy Large Hadron Collider (HE-LHC) is discussed. We apply pure leptonic and semileptonic decays for top quark pair production in the final state. Moreover, we consider systematic uncertainties of $0,3\text{\%}$ and 5%. The best limits obtained from the process $pp\to p\gamma p\to pt\overline{t}X$ on the anomalous $a_A$ and $a_V$ couplings are $\left|a_A\right|=0.0200$ and $a_V=[-0.9959;0.0003]$. Thus, our results indicate that the process $pp\to p\gamma p\to pt\overline{t}X$ is a very good perspective to probe the electric and magnetic dipole moments of the top quark at the HL-LHC and the HE-LHC.