Abstract: This paper presents direct searches for lepton flavour violation in Higgs boson decays, H -> e tau and H -> μtau, performed using data collected with the ATLAS detector at the LHC. The searches are based on a data sample of proton-proton collisions at a centre-of-mass energy root s = 13 TeV, corresponding to an integrated luminosity of 138 fb(-1). Leptonic (tau -> l nu(l)nu(tau)) and hadronic (tau -> hadrons nu(tau)) decays of the tau-lepton are considered. Two background estimation techniques are employed: the MC-template method, based on data-corrected simulation samples, and the Symmetry method, based on exploiting the symmetry between electrons and muons in the Standard Model backgrounds. No significant excess of events is observed and the results are interpreted as upper limits on lepton-flavour-violating branching ratios of the Higgs boson. The observed (expected) upper limits set on the branching ratios at 95% confidence level, B(H -> e tau) < 0.20% (0.12%) and B(H -> μtau ) < 0.18% (0.09%), are obtained with the MC-template method from a simultaneous measurement of potential H -> e tau and H -> μtau signals. The best-fit branching ratio difference, B(H -> μtau) -> B(H -> e tau), measured with the Symmetry method in the channel where the tau-lepton decays to leptons, is (0.25 0.10)%, compatible with a value of zero within 2.5 sigma.