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In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^A$ the associated Weil bundle. When $(M,{\omega }_M)$ is a Poisson manifold with $2$-form ${\omega }_M$, we construct the $2$-Poisson form ${\omega }_{M^A}^A$, prolongation on $M^A$ of the $2$-Poisson form ${\omega }_M$. We give a necessary and sufficient condition for that $M^A$ be an $A$-Poisson manifold.