This note presents direct searches for lepton flavour violation in Higgs boson decays, $H\rightarrow e\tau$ and $H\rightarrow\mu\tau$, 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 $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of 138~fb$^{-1}$. Leptonic ($\tau \to \ell \nu_\ell \nu_\tau$) and hadronic ($\tau \to \text{hadrons } \nu_\tau$) decays of the tau lepton are considered. Two background estimation techniques are exploited: 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 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 at 95\% confidence level on the branching ratios of $\mathcal{B}(H\to e\tau)<0.19\%$ ($0.11\%$) and $\mathcal{B}(H\to \mu\tau)<0.18\%$ ($0.09\%$) are obtained with the MC-template method from the simultaneous measurement of the $H \rightarrow e\tau$ and $H \rightarrow\mu\tau$ signals. The best-fit branching ratio difference, $\mathcal{B}(H\to \mu\tau)- \mathcal{B}(H\to e\tau)$, measured with the Symmetry method in the channel where the tau decays to leptons, corresponds to $0.25 \pm 0.10\%$.

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