Finding families that admit a linear $χ$-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of $2K_2$-free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of $2K_2$-free graphs, namely $(P_3\cup P_2)$-free graphs. We show that the class of $\{P_3\cup P_2,gem\}$-free graphs admits $f=2ω$ as a linear $χ$-binding function. Furthermore, we give examples to show that the optimal $χ$-binding function $f^*\geq \left\lceil\frac{5ω(G)}{4}\right\rceil$ for the class of $\{P_3\cup P_2, gem\}$-free graphs and that the $χ$-binding function $f=2ω$ is tight when $ω=2$ and $3$.