A search for heavy pseudo-scalar (A) and scalar (H) Higgs bosons decaying into a top-quark pair ($ t\overline{t} $) has been performed with 140 fb$^{−1}$ of proton-proton collision data collected by the ATLAS experiment at the Large Hadron Collider at a centre-of-mass energy of $ \sqrt{s} $ = 13 TeV. Interference effects between the signal process and Standard Model (SM) $ t\overline{t} $ production are taken into account. Final states with exactly one or exactly two electrons or muons are considered. No significant deviation from the SM prediction is observed. The results of the search are interpreted in the context of a two-Higgs-doublet model (2HDM) of type II in the alignment limit with mass-degenerate pseudo-scalar and scalar Higgs bosons (m$_{A}$ = m$_{H}$) and the hMSSM parameterisation of the minimal supersymmetric extension of the Standard Model. Ratios of the two vacuum expectation values, tan β, smaller than 3.49 (3.16) are excluded at 95% confidence level for m$_{A}$ = m$_{H}$ = 400 GeV in the 2HDM (hMSSM). Masses up to 1240 GeV are excluded for the lowest tested tan β value of 0.4 in the 2HDM. In the hMSSM, masses up to 950 GeV are excluded for tan β = 1.0. In addition, generic exclusion limits are derived separately for single scalar and pseudo-scalar states for different choices of their mass and total width.[graphic not available: see fulltext]