A search for a pseudoscalar (A) or scalar (H) heavy Higgs boson decaying into a top-quark pair () has been conducted in 20.3 fb of data collected by the ATLAS detector at the Large Hadron Collider in proton--proton collisions at a centre-of-mass energy of TeV. The analysis relies on the invariant mass spectrum of the pair in final states with an electron or muon, large missing transverse momentum, and at least four jets. Interference effects between the signal process and production in the Standard Model, which heavily distort the signal shape from a single peak to a peak-dip structure, are taken into account. Exclusion limits are derived for two resonance masses (500 and 750 GeV), as a function of the parameter tanβ in Two-Higgs-Doublet Models.