Došlić 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 $Δ$, we show $Mo(G)\leq \fracΔ{2}n^2-(1-o(1))c_Δn\log(\log(n)),$ where $c_Δ>0$ only depends on $Δ$ and the $o(1)$ term only depends on $n$. Furthermore, for integers $n_0$ and $Δ$ at least $3$, we show the existence of a $Δ$-regular graph of order $n$ at least $n_0$ with $Mo(G)\geq \fracΔ{2}n^2-c'_Δn\log(n),$ where $c'_Δ>0$ only depends on $Δ$.