  Let $T$ be a bilinear Calder\'on-Zygmund singular integral operator and $T^*$
be its corresponding truncated maximal operator. For any
$b\in\text{BMO}(\mathbb {R}^n)$ and $\vec{b}=(b_1,\ b_2)\in\text{BMO}(\mathbb
{R}^n)\times\text {BMO}(\mathbb{R}^n)$, let $T^*_{b,j}$ (j=1,2),
$T^*_{\vec{b}}\ $ be the commutators in the j-th entry and the iterated
commutators of $T^*$, respectively. In this paper, for all $1<p_1,p_2<\infty$,
$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, we show that $T^*_{b,j}$ and
$T^*_{\vec{b}}$ are compact operators from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to
$L^p(v_{\vec{w}})$, if $b,b_1,b_2\in{\rm CMO}(\mathbb{R}^n)$ and
$\vec{w}=(w_1,w_2)\in A_{\vec{p}}$, $v_{\vec{w}}=w_1^{p/p_1}w_2^{p/p_2}$. Here
${\rm CMO}(\mathbb{R}^n)$ denotes the closure of
$\mathcal{C}_c^\infty(\mathbb{R}^n)$ in the ${\rm BMO}(\mathbb{R}^n)$ topology
and $A_{\vec{p}}$ is the multiple weights class.
