A space $X$ is \emph{weakly Hurewicz} if for each sequence $(\cal U_n:n\in \Bbb N)$ of open covers of $X$, there are a dense subset $Y\subseteq X$ and finite subfamilies $\cal V_n\subseteq \cal U_n(n\in \Bbb N)$ such that for every point of $Y$ is contained in $\cup \cal V_n$ for all but finitely many $n$. In this paper, we investigate the relationship between Hurewicz spaces and weakly Hurewicz spaces, and also study topological properties of weakly Hurewicz spaces.