  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 $b \bar{b}$ pair at LHC with $\sqrt{s} =
13 $~TeV. Upper limits on the branching fractions of $H \to b\bar{s}$ and $H
\to b\bar{d}$ 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 $H
\rightarrow \gamma \gamma$ 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 $\mathcal{B}(H \rightarrow b\bar{s})$ and $\mathcal{B}(H \rightarrow
b\bar{d})$.
