We investigate the production of heavy, neutral Higgs boson resonances and their decays to top-quark top-antiquark (t t ̄) pairs at the Large Hadron Collider (LHC) at next-to-leading order (NLO) in the strong coupling of quantum chromodynamics (QCD). The NLO corrections to heavy Higgs boson production and the Higgs-QCD interference are calculated in the large mt limit with an effective K-factor rescaling. The nonresonant t t ̄ background is taken into account at NLO QCD including weak-interaction corrections. In order to consistently determine the total decay widths of the heavy Higgs bosons, we consider for definiteness the type-II two-Higgs-doublet extension of the standard model and choose three parameter scenarios that entail two heavy neutral Higgs bosons with masses above the t t ̄ threshold and unsuppressed Yukawa couplings to top quarks. For these three scenarios we compute, for the LHC operating at 13 TeV, the t t ̄ cross section and the distributions of the t t ̄ invariant mass, of the transverse top-quark momentum and rapidity, and of the cosine of the Collins-Soper angle with and without the two heavy Higgs resonances. For selected Mt t ̄ bins we estimate the significances for detecting a heavy Higgs signal in the t t ̄ dileptonic and lepton plus jets decay channels.