  Let $T$ be a bilinear Calder\'{o}n-Zygmund singular integral operator and
$T_*$ be its corresponding truncated maximal operator. The commutators in the
$i$-$th$ entry and the iterated commutators of $T_*$ are defined by $$
T_{\ast,b,1}(f,g)(x)=\sup_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}K(x,y,z)(b(y)-b(x))f(y)g(z)dydz\bigg|,
$$
$$T_{\ast,b,2}(f,g)(x)=\sup_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}K(x,y,z)(b(z)-b(x))f(y)g(z)dydz\bigg|,$$
\begin{align*}
T_{\ast,(b_1,b_2)}(f,g)(x)=\sup\limits_{\delta>0}\bigg|\iint_{|x-y|+|x-z|>\delta}
K(x,y,z)(b_1(y)-b_1(x))(b_2(z)-b_2(x))f(y)g(z)dydz\bigg|. \end{align*} In this
paper, the compactness of the commutators $T_{\ast,b,1}$, $T_{\ast,b,2}$ and
$T_{\ast,(b_1,b_2)}$ on $L^r(\mathbb{R}^n))$ is established.
