Abstract A search for the associated production of a heavy resonance with a top-quark or a top-antitop-quark pair, and decaying into a $$t{\bar{t}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mover> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> pair is presented. The search uses the data recorded by the ATLAS detector in pp collisions at $$\sqrt{s}= 13$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> </mml:mrow> </mml:math> $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TeV</mml:mtext> </mml:math> at the Large Hadron Collider during the years 2015–2018, corresponding to an integrated luminosity of 139 $$\text {fb}^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>fb</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . 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 $$R=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The invariant mass spectrum of the two top quark candidates is used to search for a resonance signal in the range of 1.0 $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TeV</mml:mtext> </mml:math> to 3.2 $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TeV</mml:mtext> </mml:math> . 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 $$Z^\prime $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> boson based on a simplified model, for $$Z^\prime $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> mass between 1.0 $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TeV</mml:mtext> </mml:math> and 3.0 $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TeV</mml:mtext> </mml:math> . The observed (expected) limits range from 21 (14) fb to 119 (86) fb depending on the choice of model parameters.