Let us check one quadrant for when squares with $2$ ($3,4,5$) written in them are surrounded by squares with $3$ ($4,5,6$) written in them.
$\begin{tabular} {|c|c|c|c|c|c|c|c|c|c|} \cline{1-10}
O & 3 & 2 & 3 & {\bf 2} & 3 & 4 & 5 & {\bf 4} & 5 \\ \cline{1-10}
3 & 2 & 1 & 2 & 3 & 4 & {\bf 3} & 4 & 5 & 6 \\ \cline{1-10}
2 & 1 & 4 & 3 & {\bf 2} & 3 & 4 & 5 & {\bf 4} & 5 \\ \cline{1-10}
3 & 2 & 3 & {\bf 2} & 3 & 4 & {\bf 3} & 4 & 5 & 6 \\ \cline{1-10}
{\bf 2} & 3 & {\bf 2} & 3 & 4 & {\bf 3} & 4 & 5 & {\bf 4} & 5 \\ \cline{1-10}
3 & 4 & 3 & 4 & {\bf 3} & 4 & 5 & {\bf 4} & 5 & 6 \\ \cline{1-10}
4 & {\bf 3} & 4 & {\bf 3} & 4 & 5 & {\bf 4} & 5 & 6 & {\bf 5} \\ \cline{1-10}
5 & 4 & 5 & 4 & 5 & {\bf 4} & 5 & 6 & {\bf 5} & 6 \\ \cline{1-10}
{\bf 4} & 5 & {\bf 4} & 5 & {\bf 4} & 5 & 6 & {\bf 5} & 6 & 7 \\ \cline{1-10}
5 & 6 & 5 & 6 & 5 & 6 & {\bf 5} & 6 & 7 & {\bf 6} \\ \cline{1-10}
\end{tabular}$
For squares containing $k\geq 3$ surrounded by squares with $k+1$ written in them, there is a structure of squares containing $k+1$ surrounded by squares with $k$ written in them, just interior to the former.
The pattern is visible, and extends to $16$ such squares for $2$, $32$ such squares for $4$, $\ldots,$ $800$ such squares for $100$.