  The class of $2K_2$-free graphs have been well studied in various contexts in
the past. It is known that the class of $\{2K_2,2K_1+K_p\}$-free graphs and
$\{2K_2,(K_1\cup K_2)+K_p\}$-free graphs admits a linear $\chi$-binding
function. In this paper, we study the classes of $(P_3\cup P_2)$-free graphs
which is a superclass of $2K_2$-free graphs. We show that $\{P_3\cup
P_2,2K_1+K_p\}$-free graphs and $\{P_3\cup P_2,(K_1\cup K_2)+K_p\}$-free graphs
also admits linear $\chi$-binding functions. In addition, we give tight
chromatic bounds for $\{P_3\cup P_2,HVN\}$-free graphs and $\{P_3\cup
P_2,diamond\}$-free graphs and it can be seen that the latter is an improvement
of the existing bound given by A. P. Bharathi and S. A. Choudum [Colouring of
$(P_3\cup P_2)$-free graphs, Graphs and Combinatorics 34 (2018), 97-107].
