  A record in a permutation is a maximum or a minimum, from the left or from
the right. The entries of a permutation can be partitioned into two types: the
ones that are records are called external points, the others are called
internal points. Permutations without internal points have been studied under
the name of square permutations. Here, we explore permutations with a fixed
number of internals points, called almost square permutations. Unlike with
square permutations, a precise enumeration for the total number of almost
square permutations of size $n+k$ with exactly $k$ internal points is not
known. However, using a probabilistic approach, we are able to determine the
asymptotic enumeration. This allows us to describe the permuton limit of almost
square permutations with $k$ internal points, both when $k$ is fixed and when
$k$ tends to infinity along a negligible sequence with respect to the size of
the permutation. Finally, we show that our techniques are quite general by
studying the set of $321$-avoiding permutations of size $n$ with exactly $k$
additional internal points ($k$ fixed). In this case we obtain an interesting
asymptotic enumeration in terms of the Brownian excursion area. As a
consequence, we show that the points of a uniform permutation in this set
concentrate on the diagonal and the fluctuations of these points converge in
distribution to a biased Brownian excursion.
