  Considering the MSSM with the CKM matrix as the only source of flavour
violation and heavy supersymmetric particles at large $\tan\beta$, we analyze
the correlation between {\it the increase} of the rates of the decays
$B^0_{s,d}\to \mu^+\mu^-$ and {\it the suppression} of $\Delta M_s$, that are
caused by the enhanced flavour changing neutral Higgs couplings to down-type
quarks. We give analytic formulae for the neutral and charged Higgs couplings
to quarks including large $\tan\beta$ resummed corrections in the $SU(2)\times
U(1)$ limit and comment briefly on the accuracy of this approximation. For
$0.8\le (\Delta M_s)^{\rm exp}/(\Delta M_s)^{\rm SM}\le 0.95$ we find $6\cdot
10^{-7}\ge BR(B^0_s\ra \mu^+\mu^-)^{\rm max} \ge 4\cdot 10^{-8}$ and $1.4\cdot
10^{-8}\ge BR(B^0_d\ra \mu^+\mu^-)^{\rm max}\ge 1\cdot 10^{-9}$. For $(\Delta
M_s)^{\rm exp} \ge (\Delta M_s)^{\rm SM}$ substantial enhancements of
$B^0_{s,d}\ra\mu^+\mu^-$ relative to the expectations based on the Standard
Model are excluded. With $(\Delta M_s)^{\rm exp}>15.0/$ps a conservative
analysis of $(\Delta M_s)^{\rm SM}$ gives $BR(B^0_s\ra
\mu^+\mu^-)\simlt1.2\cdot10^{-6}$ and $BR(B^0_d\ra
\mu^+\mu^-)\simlt3\cdot10^{-8}$. However, we point out that in the less likely
scenario in which the squark mixing is so large that the neutral Higgs
contributions dominate $\Delta M_s$, the rates for $B^0_{s,d}\to \mu^+\mu^-$
increase with increasing $\Delta M_s$ and the bounds in question are weaker.
Violation of all these correlations and bounds would indicate new sources of
flavour violation.
