  Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as
$Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of
$G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a
smaller distance in $G$ to $u$ than to $v$. For a graph $G$ of order $n$ and
maximum degree at most $\Delta$, we show $Mo(G)\leq
\frac{\Delta}{2}n^2-(1-o(1))c_{\Delta}n\log(\log(n)),$ where $c_{\Delta}>0$
only depends on $\Delta$ and the $o(1)$ term only depends on $n$. Furthermore,
for integers $n_0$ and $\Delta$ at least $3$, we show the existence of a
$\Delta$-regular graph of order $n$ at least $n_0$ with $Mo(G)\geq
\frac{\Delta}{2}n^2-c'_{\Delta}n\log(n),$ where $c'_{\Delta}>0$ only depends on
$\Delta$.
