The voter model with stirring is a variant of the classical voter model on $\mathbb{Z}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf v \ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d \ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{ μ_{α,\mathsf{v}}: α\in [0,1] \}$, where~$α$ is the density of voters with opinion~1. Sampling a configuration~$ξ$ from~$μ_{α, \mathsf v}$, we study~$ξ$ as a site percolation model on~$\mathbb{Z}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$ξ$. Letting~$α_c(\mathsf v)$ be the supremum of all the values of~$α$ for which percolation does not occur~$μ_{α, \mathsf v}$-a.s., we prove that $α_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf v$ large enough, the model exhibits a non-trivial phase transition in~$α$.