  Let $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have
mean value zero, $T_{\Omega}$ be the homogeneous singular integral operator
with kernel $\frac{\Omega(x)}{|x|^d}$ and $T_{\Omega}^*$ be the maximal
operator associated to $T_{\Omega}$. In this paper, the authors prove that if
$\Omega\in L^{\infty}(S^{d-1})$, then for all $r\in (1,\,\infty)$,
$T_{\Omega}^*$ enjoys a $(L^\Phi,\,L^r)$ bilinear sparse domination with bound
$Cr'\|\Omega\|_{L^{\infty}(S^{d-1})}$, where $\Phi(t)=t\log\log ({\rm e}^2+t)$.
