  Navarro has conjectured a necessary and sufficient condition for a finite
group $G$ to have a self-normalising Sylow $2$-subgroup, which is given in
terms of the ordinary irreducible characters of $G$. The first-named author has
reduced the proof of this conjecture to showing that certain related statements
hold when $G$ is quasisimple. In this article we show that these conditions are
satisfied when $G/Z(G)$ is $\mathrm{PSL}_n(q)$, $\mathrm{PSU}_n(q)$, or a
simple group of Lie type defined over a finite field of characteristic $2$.
