A space $X$ is \emph{almost Menger (weakly Menger)} if for each sequence $(\U_n:n\in\mathbb N)$ of open covers of $X$ there exists a sequence $(\mathcal V_n:n\in\mathbb N)$ such that for every $n\in\mathbb N$, $\mathcal V_n$ is a finite subset of $\U_n$ and $\bigcup_{n\in\mathbb N}\bigcup\big\{\overline{V}:V\in\mathcal V_n\big\}=X$ (respectively, $\overline{\bigcup_{n\in\mathbb N}\bigcup\{V:V\in\mathcal V_n\}}=X$). We investigate the relationships among almost Menger spaces, weakly Menger spaces and Menger spaces, and also study topological properties of almost Menger spaces and weakly Menger spaces.