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

Linear $\chi$-binding functions for $\{P_3\cup P_2, gem\}$-free graphs