Motivated by the Arhangel’skii [2] “s-Lindel¨of cardinal function” and Koˇcinac, Konca, and Singh [15] set-star covering properties, we introduce the setstar-C-Menger property. A space X is said to have the set-star-C-Menger property if for each nonempty subset A of X and each sequence (Un : n ∈ N) of families of open subsets of X such that A ⊂ ∪Un for each n ∈ N, there is a sequence (Kn : n ∈ N) of countably compact subsets of X such that A ⊂ S n∈N St(Kn, Un). In this paper, we investigate the relationship between the set-star-C-Menger and other related properties and study the topological properties of the set-star-C-Menger property.