A space $X$ is said to be neighborhood star-Menger if for every sequence $(\mathcal{U}_n:n\in\mathbb{N})$ of open covers of $X$ one can choose finite $A_n\subseteq X$, $n\in\N$ such that for every open $O_n\supseteq A_n$, $n\in\N$, $\{\operatorname{St}(O_n,\mathcal{U}_n):n\in\mathbb{N}\}$ is an open cover of $X$. We investigate the relationship between neighborhood star-Menger spaces and related spaces, and study the topological properties of neighborhood star-Menger spaces.