A measurement is presented of $t\bar{t}$ production in the invariant-mass region near the pair production threshold, $m_{t\bar{t}} \sim 345\,\mathrm{GeV}$, in final states with two charged leptons and multiple jets. The measurement is based on $140\,\mathrm{fb}^{-1}$ of proton–proton collision data collected at $\sqrt{s} = 13\,\mathrm{TeV}$ with the ATLAS detector. The data are compared to two models of $t\bar{t}$ production: a baseline model including only perturbative QCD (pQCD) predictions for the hard process, and an extended model that, in addition to the pQCD predictions, incorporates Monte Carlo simulations of color-singlet quasi-bound-state formation near the $t\bar{t}$ threshold. An excess of events is observed over the baseline pQCD prediction, with an observed significance of $7.7$ standard deviations. This excess is consistent with the formation of colour-singlet, $S$-wave, quasi-bound $t\bar{t}$ states as predicted by non-relativistic QCD, and corresponds to an measured cross-section of $9.0 \pm 1.3\,\mathrm{pb}$.