The concept of $χ$-binding functions for classes of free graphs has been extensively studied in the past. In this paper, we improve the existing $χ$-binding function for $\{2K_2, K_1 + C_4\}$-free graphs. Also, we find a linear $χ$-binding function for $\{2K_2, K_2+P_4\}$-free graphs. In addition, we give alternative proofs for the $χ$-binding function of $\{2K_2,gem\}$-free graphs, $\{2K_2,HVN\}$-free graphs and $\{2K_2, K_5-e\}$-free graphs. Finally, for $p\geq3$, we find polynomial $χ$-binding functions for $\{pK_2, H\}$-free graphs where $H\in \{gem, diamond, K_2+P_4, HVN, K_5-e, butterfly, gem^+, dart, K_1 + C_4, C_4, \overline{P_5}\}$.