A search for heavy neutral Higgs bosons produced in association with one or two $b$-quarks and decaying to $b$-quark pairs is presented using 27.8 fb$^{-1}$ of $\sqrt{s}=13$~TeV proton--proton collision data recorded by the ATLAS detector at the Large Hadron Collider during 2015 and 2016. No evidence of a signal is found. Upper limits on the heavy neutral Higgs boson production cross-section times its branching ratio to $b\bar{b}$ are set, ranging from 4.0 to 0.6 pb at 95\% confidence level over a Higgs boson mass range of 450 to 1400~GeV. Results are interpreted within the Two Higgs Doublet Model and the Minimal Supersymmetric Standard Model.

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