A hereditary class $\cal G$ of graphs is {\em $χ$-bounded} if there is a {\em $χ$-binding function}, say $f$, such that $χ(G)\le f(ω(G))$ for every $G\in\cal G$, where $χ(G)(ω(G))$ denotes the chromatic (clique) number of $G$. It is known that for every $(P_3\cup P_2)$-free graph $G$, $χ(G)\le \frac{1}{6}ω(G)(ω(G)+1)(ω(G)+2)$ \cite{BA18}, and the class of $(2K_2, 3K_1)$-free graphs does not admit a linear $χ$-binding function\cite{BBS19}. In this paper, we prove that (\romannumeral 1) $χ(G)\le2ω(G)$ if $G$ is ($P_3\cup P_2$, kite)-free, (\romannumeral 2) $χ(G)\leω^2(G)$ if $G$ is ($P_3\cup P_2$, hammer)-free, (\romannumeral 3) $χ(G)\le\frac{3ω^2(G)+ω(G)}{2}$ if $G$ is ($P_3\cup P_2, C_5$)-free. Furthermore, we also discuss $χ$-binding functions for $(P_3\cup P_2, K_4)$-free graphs.