  The $r$-neighbor bootstrap percolation is a graph infection process based on
the update rule by which a vertex with $r$ infected neighbors becomes infected.
We say that an initial set of infected vertices propagates if all vertices of a
graph $G$ are eventually infected, and the minimum cardinality of such a set in
$G$ is called the $r$-bootstrap percolation number, $m(G,r)$, of $G$. In this
paper, we study percolating sets in direct products of graphs. While in general
graphs there is no non-trivial upper bound on $m(G\times H,r)$, we prove
several upper bounds under the assumption $\delta(G)\ge r$. We also
characterize the connected graphs $G$ and $H$ with minimum degree $2$ that
satisfy $m(G \times H, 2) = \frac{|V(G \times H)|}{2}$. In addition, we
determine the exact values of $m(P_n \times P_m, 2)$, which are $m+n-1$ if $m$
and $n$ are of different parities, and $m+n$ otherwise.
