  The observations in the $B_{s}$ sector suggest the existence of some new
physics contribution to the $B_S-\bar{B}_s$ mixing. We study the implications
of a hypothesis that this contribution is generated by the Higgs induced
flavour changing neutral currents. We concentrate on the specific $b\to s$
transition which is described by two complex FCNC parameters $F_{23}$ and
$F_{32}$ and parameters in the Higgs sector. Model independent constraints on
these parameters are derived from the $B_S-\bar{B}_s$ mixing and are used to
predict the branching ratios for $\bar{B}_s\to \mu^+\mu^-$ and $\bar{B}_d\to
\bar{K}\mu^+\mu^-$ numerically by considering general variations in the Higgs
parameters assuming that Higgs sector conserves CP. Taking the results on
$B_{s}^0-\bar{B}_{s}^0$ mixing derived by the global analysis of UTfit group as
a guide we present the general constraints on $F_{23}^*F_{32}$ in terms of the
pseudo-scalar mass $M_A$. The former is required to be in the range $\sim (1-5)
\times 10^{-11} M_A^2 {\rm GeV}^{-2}$ if the Higgs induced FCNC represent the
dominant source of new physics. The phases of these couplings can account for
the large CP violating phase in the $B_{s}^0-\bar{B}_{s}^0$ mixing except when
$F_{23}=F_{32}$. The Higgs contribution to $\bar{B}_s\to \mu^+\mu^-$ branching
ratio can be large, close to the present limit while it remains close to the
standard model value in case of the process $\bar{B}_d\to \bar{K}\mu^+\mu^-$
for all the models under study. We identify and discuss various specific
examples which can naturally lead to suppressed FCNC in the $K^0-\bar{K}^0$
mixing allowing at the same time the required values for $F_{23}^*$ and
$F_{32}$.
