A topological space $X$ is called $G_{\delta}$-selectively (resp., $SG_{\delta}$-selectively) separable if for every sequence $(D_n:n\in\omega)$ of dense $G_\delta$ subsets of $X$, one can pick finite subsets $F_n\subset D_n$ such that $\bigcup_{n\in\omega}F_n$ is dense (resp., dense and $G_\delta$). In this paper we introduce and study these kinds of spaces.