A search for heavy neutral Higgs bosons is performed using the LHC Run 2 data, corresponding to an integrated luminosity of 139  fb-1 of proton-proton collisions at s=13  TeV recorded with the ATLAS detector. The search for heavy resonances is performed over the mass range 0.2–2.5 TeV for the τ+τ- decay with at least one τ-lepton decaying into final states with hadrons. The data are in good agreement with the background prediction of the standard model. In the Mh125 scenario of the minimal supersymmetric standard model, values of tanβ>8 and tanβ>21 are excluded at the 95% confidence level for neutral Higgs boson masses of 1.0 and 1.5 TeV, respectively, where tanβ is the ratio of the vacuum expectation values of the two Higgs doublets.