A search for the associated production of a heavy resonance with a top-quark or a top-antitop-quark pair, and decaying into a tt¯ pair is presented. The search uses the data recorded by the ATLAS detector in pp collisions at s=13 TeV at the Large Hadron Collider during the years 2015–2018, corresponding to an integrated luminosity of 139 fb-1. Events containing exactly one electron or muon are selected. The two hadronically decaying top quarks from the resonance decay are reconstructed using jets clustered with a large radius parameter of R=1. The invariant mass spectrum of the two top quark candidates is used to search for a resonance signal in the range of 1.0 TeV to 3.2 TeV. The presence of a signal is examined using an approach with minimal model dependence followed by a model-dependent interpretation. No significant excess is observed over the background expectation. Upper limits on the production cross section times branching ratio at 95% confidence level are provided for a heavy Z′ boson based on a simplified model, for Z′ mass between 1.0 TeV and 3.0 TeV. The observed (expected) limits range from 21 (14) fb to 119 (86) fb depending on the choice of model parameters.