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.