  Let $r$ be a positive integer, and let $A$ be a nonempty finite set of at
least two integers. We let $\tilde{C}_r(A)$ denote the {\em asymptotic
$r$-covering number} of $A$, that is, the smallest integer value of $l$ for
which, for all sufficiently large positive integers $h$, the $rh$-fold sumset
of $A$ is contained in at most $l$ translates of the $h$-fold sumset of $A$.
Nathanson proved that $\tilde{C}_r(A)$ is always at most $r+1$; here we extend
this result to prove that $\tilde{C}_r(A)$ is always at least $r$, and
determine all sets $A$ for which $\tilde{C}_r(A)=r$.
