We investigate when the product of two smooth manifolds admits a weakly Lagrangian embedding. Prove that, if $M^m$ and $N^n$ are smooth manifolds such that M admits a weakly Lagrangian embedding into ${mathbb}C^m$ whose normal bundle has a nowhere vanishing section and N admits a weakly Lagrangian immersion into ${mathbb}C^n$, then $M imes N$ admits a weakly Lagrangian embedding into ${mathbb}C^{m+n}$. As a corollary, we obtain that $S^m { imes} S^n$ admits a weakly Lagrangian embedding into ${mathbb}C^{m+n}$ if n=1,3. We investigate the problem of whether $S^m{ imes}S^n$ in general admits a weakly Lagrangian embedding into ${mathbb} C^{m+n}$