A hereditary class $\mathcal{G}$ of graphs is $��$-bounded if there is a $��$-binding function, say $f$ such that $��(G) \leq f(��(G))$, for every $G \in \cal{G}$, where $��(G)$ ($��(G)$) denote the chromatic (clique) number of $G$. It is known that for every $2K_2$-free graph $G$, $��(G) \leq \binom{��(G)+1}{2}$, and the class of ($2K_2, 3K_1$)-free graphs does not admit a linear $��$-binding function. In this paper, we are interested in classes of $2K_2$-free graphs that admit a linear $��$-binding function. We show that the class of ($2K_2, H$)-free graphs, where $H\in \{K_1+P_4, K_1+C_4, \overline{P_2\cup P_3}, HVN, K_5-e, K_5\}$ admits a linear $��$-binding function. Also, we show that some superclasses of $2K_2$-free graphs are $��$-bounded.

Revised version