A bstract A search for flavour-changing neutral current decays of a top quark into an up-type quark ( q = u , c ) and the Standard Model Higgs boson, t → Hq , is presented. The search is based on a dataset of pp collisions at $$ \sqrt{s} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> </mml:math> = 13 TeV recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider and corresponding to an integrated luminosity of 36.1 fb −1 . Two complementary analyses are performed to search for top-quark pair events in which one top quark decays into Wb and the other top quark decays into Hq , and target the $$ H\to b\overline{b} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> <mml:mo>→</mml:mo> <mml:mi>b</mml:mi> <mml:mover> <mml:mi>b</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> and H → τ + τ − decay modes, respectively. The high multiplicity of b -quark jets, or the presence of hadronically decaying τ -leptons, is exploited in the two analyses respectively. Multivariate techniques are used to separate the signal from the background, which is dominated by top-quark pair production. No significant excess of events above the background expectation is found, and 95% CL upper limits on the t → Hq branching ratios are derived. The combination of these searches with ATLAS searches in diphoton and multilepton final states yields observed (expected) 95% CL upper limits on the t → Hc and t → Hu branching ratios of 1.1 × 10 −3 (8.3 × 10 −4 ) and 1.2 × 10 −3 (8.3 × 10 −4 ), respectively. The corresponding combined observed (expected) upper limits on the | λ tcH | and | λ tuH | couplings are 0.064 (0.055) and 0.066 (0.055), respectively.