  We investigate hermitian Yang--Mills connections for pullback vector bundles
on blow-ups of K\"ahler manifolds along submanifolds. Under some mild
asumptions on the graded object of a simple and semi-stable vector bundle, we
provide a necessary and sufficent numerical criterion for the pullback bundle
to admit a sequence of hermitian Yang--Mills connections for polarisations that
make the exceptional divisor sufficiently small, and show that those
connections converge to the pulled back hermitian Yang-Mills connection of the
graded object.
