  Let $X$ be a K3 surface which doubly covers an Enriques surface $S$. If
$n\in\mathbb{N}$ is an odd number, then the Hilbert scheme of $n$-points
$X^{[n]}$ admits a natural quotient $S_{[n]}$. This quotient is an Enriques
manifold in the sense of Oguiso and Schr\"oer. In this paper we construct slope
stable sheaves on $S_{[n]}$ and study some of their properties.
