This paper presents a search for new heavy particles decaying into a pair of top quarks using 139 fb of proton-proton collision data recorded at a centre-of-mass energy of s = 13 TeV with the ATLAS detector at the Large Hadron Collider. The search is performed using events consistent with pair production of high-transverse-momentum top quarks and their subsequent decays into the fully hadronic final states. The analysis is optimized for resonances decaying into a tt¯ pair with mass above 1.4 TeV, exploiting a dedicated multivariate technique with jet substructure to identify hadronically decaying top quarks using large-radius jets and evaluating the background expectation from data. No significant deviation from the background prediction is observed. Limits are set on the production cross-section times branching fraction for the new Z′ boson in a topcolor-assisted-technicolor model. The Z′ boson masses below 3.9 and 4.7 TeV are excluded at 95% confidence level for the decay widths of 1% and 3%, respectively. [Figure not available: see fulltext.]