Let $I$ be a matroidal ideal of degree $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let $\astab(I)$ and $\dstab(I)$ be the smallest integers $m$ and $n$, for which $\Ass(I^m)$ and $\depth(I^n)$ stabilize, respectively. In this paper, we show that $\astab(I)=1$ if and only if $\dstab(I)=1$. Moreover, we prove that if $d=3$, then $\astab(I)=\dstab(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then $\astab(I)=\dstab(I)=\lceil\frac{n-1}{n-d}\rceil$.
