We consider the Higgs boson anomalous FCNC interactions with u, c, d, s and b quarks using the effective field theory framework. Constraints on anomalous couplings are derived from experimental results on Higgs boson production with subsequent decay into $b\overline{b}$ pair at LHC with $\sqrt{s}=13$ TeV. Upper limits on the branching fractions of $H\to b\overline{s}$ and $H\to b\overline{d}$ are set by performing a realistic detector simulation and accurately reproducing analysis selections of the CMS Higgs boson measurement in the four-lepton final state at $\sqrt{s}=13$ TeV. The searches are projected into operation conditions of HL-LHC. Sensitivity at FCC-hh to anomalous FCNC interactions is studied based on Higgs boson production with $H\to \gamma \gamma$ decay channel. It is shown that at FCC-hh machine one can expect to set the upper limits of the order of ${10}^{-2}$ at 95% CL for $B(H\to b\overline{s})$ and $B(H\to b\overline{d})$.