Charged Higgs bosons produced either in top-quark decays or in association with a top quark, subsequently decaying via H±→τ±ντ, are searched for in 140 fb−1 of proton-proton collision data at s=13 TeV recorded with the ATLAS detector. Depending on whether the top quark is produced together with the H± decays hadronically or semileptonically, the search targets τ+jets or τ+lepton final states, in both cases with a τ-lepton decaying into a neutrino and hadrons. No significant excess over the Standard Model background expectation is observed. For the mass range of 80≤mH±≤3000 GeV, upper limits at 95% confidence level are set on the production cross section of the charged Higgs boson times the branching fraction B(H±→τ±ντ) in the range 4.5 pb–0.4 fb. In the mass range 80–160 GeV, assuming the Standard Model cross section for tt¯ production, this corresponds to upper limits between 0.27% and 0.02% on B(t→bH±)×B(H±→τ±ντ).