  We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have
the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are
sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact
in the projective tensor product $X \widehat{\otimes}_\pi Y$, then either $W_1$
or $W_2$ is relatively norm compact. For instance, such a property holds for
the pair $(\ell_p,\ell_q)$ if $1<p,q<\infty$ satisfy $1/p+1/q\geq 1$. Other
examples are given that allow us to provide alternative proofs to some results
on multiplication operators due to Saksman and Tylli. We also revisit, with
more direct proofs, some known results about the embeddability of $\ell_1$ into
$X \widehat{\otimes}_\pi Y$ for arbitrary Banach spaces $X$ and $Y$, in
connection with the compactness of all operators from $X$ to $Y^*$.
