We study the effects of T-odd interactions of top-quark via the pair production of the top-quark in the semileptonic detection modes at the Large Hadron Collider by means of the T-odd observables constructed through the momenta of the observed decay products of the top (and anti-top)-quark for a wide range of CP-violating scale $\Lambda$. Estimates on sensitivities of the coupling strength of such interactions for 13 TeV LHC energy with $\int Ldt=36.1\text{f}{\text{b}}^{-1}$, $140\text{f}{\text{b}}^{-1}$ and for HL-LHC with 14 TeV energy with integrated luminosities of $0.3\text{a}{\text{b}}^{-1}$, $1\text{a}{\text{b}}^{-1}$, $2\text{a}{\text{b}}^{-1}$, and $3\text{a}{\text{b}}^{-1}$ are also presented for $\Lambda$ ranging between $M_W$ and 2 TeV.