Let $G$ be a graph. We use $χ(G)$ and $ω(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices. A family of graphs $\mathcal{G}$ is said to be {\it$χ$-bounded} if there exists some function $f$ such that $χ(G)\leq f(ω(G))$ for every $G\in\mathcal{G}$. In this paper, we show that the family of $(P_5, K_5-e)$-free graphs is $χ$-bounded by a linear function: $χ(G)\leq \max\{13,ω(G)+1\}$.