  We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost
simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that
$s\leqslant2$ for such digraphs, which provides the first step in determining
an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.
