It is proved that a Banach space $X$ is a subspace of a weakly compactly generated Banach space if and only if, for every $\varepsilon\,{>}\,0$, $X$ can be covered by a countable collection of bounded closed convex symmetric sets where the weak$^*$ closure in $X^{**}$ of each of them lies within the distance $\varepsilon$ from $X$. A new short functional-analytic proof of the known result that a continuous image of an Eberlein compact is Eberlein is given as a corollary.