A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prove that the $k$-forested choosability of a graph with maximum degree $\Delta\geq k\geq 4$ is at most $\left\lceil\frac{\Delta}{k-1}\right\rceil+1$, $\left\lceil\frac{\Delta}{k-1}\right\rceil+2$ or $\left\lceil\frac{\Delta}{k-1}\right\rceil+3$ if its maximum average degree is less than $\frac{12}{5}$, $\frac{8}{3}$ or $3$, respectively.