  The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest
integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any
two vertices receiving color $i$ are at distance at least $i+1$. In this short
note we present upper and lower bound for the packing chromatic number of the
lexicographic product $G\circ H$ of graphs $G$ and $H$. Both bounds coincide in
many cases. In particular this happens if $|V(H)|-\alpha(H)\geq {\rm
diam}(G)-1$, where $\alpha(G)$ denotes the independence number of $G$.
