In the weighted theory of multilinear operators, the weights class which usually has been considered is the product of $$A_p$$ weights. However, it is known that $$\prod _{k=1}^2A_{p_k}({\mathbb {R}}^n)\varsubsetneq A_{\vec {p}}({\mathbb {R}}^{2n})$$, and $$\vec {w}=(w_1,\,w_2)\in A_{\vec {p}}({\mathbb {R}}^{2n})$$ does not imply that $$w_k\in L^1_{\mathrm{loc}}({\mathbb {R}}^n)$$ for $$k=1,\,2$$. Therefore, it is very interesting to study the weighted theory of multilinear operators with the weights in $$A_{\vec {p}}({\mathbb {R}}^{2n})$$. In this paper, we consider the weights class $$A_{\vec {p}/\vec {r}}({\mathbb {R}}^{2n})$$, which is more general than $$A_{\vec {p}}({\mathbb {R}}^{2n})$$. If $$\vec {w}=(w_1,\,w_2)\in A_{\vec {p}/\vec {r}}({\mathbb {R}}^{2n})$$, we show that the bilinear Fourier multiplier operator $$T_{\sigma }$$ is bounded from $$L^{p_1}(w_1)\times L^{p_2}(w_2)$$ to $$L^p(\nu _{\vec {w}})$$ when the symbol $$\sigma $$ satisfies the Sobolev regularity that $$\sup _{\kappa \in {\mathbb {Z}}}\Vert \sigma _k\Vert _{W^{s_1,s_2}({\mathbb {R}}^{2n})} <\infty $$ with $$ s_1,s_2\in (\frac{n}{2},\,n].$$