Let $X$ be an algebraic $K3$ surface. Fix an ample divisor $H$ on $X,L\in Pic(X)$ and $c_2\in\mathbb{Z}$. Let $M_H(r; L, c_2)$ be the moduli space of rank $r,H$-stable vector bundles $E$ over $X$ with det($E) = L$ and $c_2(E) = c_2$. The goal of this paper is to determine invariants ($r; c_1, c_2$) for which $M_H(r; L, c_2)$ is birational to some Hilbert scheme $Hilb^l(X)$.