The Lund jet plane (LJP) is measured for the first time in $t\bar{t}$ events, using 140 $\textrm{fb}^{-1}$ of $\sqrt{s} = 13$ TeV pp collision data collected with the ATLAS detector at the LHC. The LJP is a two-dimensional observable of the sub-structure of hadronic jets that acts as a proxy for the kinematics of parton showers and hadron formation. The observable is constructed from charged particles and is measured for $R=1.0$ anti-$k_t$ jets with transverse momentum above 350 GeV containing the full decay products of either a top quark or a daughter W boson. The other top quark in the event is identified from its decay into a b-quark, an electron or a muon and a neutrino. The measurement is corrected for detector effects and compared with a range of Monte Carlo predictions sensitive to different aspects of the hadronic decays of the heavy particles. In the W-boson-initiated jets, all the predictions are incompatible with the measurement. In the top quark initiated jets, disagreement with all predictions is observed in smaller subregions of the plane, and with a subset of the predictions across the fiducial plane. The measurement could be used to improve the tuning of Monte Carlo generators, for better modelling of hadronic decays of heavy quarks and bosons, or to improve the performance of jet taggers.