Abstract A search for a heavy neutral Higgs boson, A , decaying into a Z boson and another heavy Higgs boson, H , is performed using a data sample corresponding to an integrated luminosity of 139 fb $$^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow/><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> from proton–proton collisions at $$\sqrt{s} = 13$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>13</mml:mn></mml:mrow></mml:math> $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>TeV</mml:mtext></mml:math> recorded by the ATLAS detector at the LHC. The search considers the Z boson decaying into electrons or muons and the H boson into a pair of b -quarks or W bosons. The mass range considered is 230–800 $$\text {GeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>GeV</mml:mtext></mml:math> for the A boson and 130–700 $$\text {GeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>GeV</mml:mtext></mml:math> for the H boson. The data are in good agreement with the background predicted by the Standard Model, and therefore 95% confidence-level upper limits for $$\sigma \times B(A\rightarrow ZH)\times B(H\rightarrow bb \; \text {or} \; H \rightarrow WW)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>×</mml:mo><mml:mi>B</mml:mi><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>→</mml:mo><mml:mi>Z</mml:mi><mml:mi>H</mml:mi><mml:mo>)</mml:mo><mml:mo>×</mml:mo><mml:mi>B</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>b</mml:mi><mml:mi>b</mml:mi><mml:mspace/><mml:mtext>or</mml:mtext><mml:mspace/><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>W</mml:mi><mml:mi>W</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> are set. The upper limits are in the range 0.0062–0.380 pb for the $$H\rightarrow bb$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>b</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:math> channel and in the range 0.023–8.9 pb for the $$H\rightarrow WW$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>W</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:math> channel. An interpretation of the results in the context of two-Higgs-doublet models is also given.