Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setminus \lbrace c\rbrace$ is not an asymptotic complement $\forall c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in $\mathbb{Z}$ and deal with a problem of Nathanson on their existence and their inexistence.