  We study bilinear rough singular integral operators $\mathcal{L}_{\Omega}$
associated with a function $\Omega$ on the sphere $\mathbb{S}^{2n-1}$.
  In the recent work of Grafakos, He, and Slav\'ikov\'a (Math. Ann. 376:
431-455, 2020), they showed that $\mathcal{L}_{\Omega}$ is bounded from
$L^2\times L^2$ to $L^1$, provided that $\Omega\in L^q(\mathbb{S}^{2n-1})$ for
$4/3<q\le \infty$ with mean value zero. We generalize their result to the
boundedness for all Banach points. We actually prove $L^{p_1}\times L^{p_2}\to
L^p$ estimates for $\mathcal{L}_{\Omega}$ under the assumption
  $$\Omega\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for
}~\max{\Big(\;\frac{4}{3}\;,\; \frac{p}{2p-1} \;\Big)<q\le \infty}$$ where
$1<p_1,p_2<\infty$ with $1/p=1/p_1+1/p_2$. Our result improves that of
Grafakos, He, and Honz\'ik (Adv. Math. 326: 54-78, 2018), in which the more
restrictive condition $\Omega\in L^{\infty}(\mathbb{S}^{2n-1})$ is required for
the $L^{p_1}\times L^{p_2}\to L^p$ boundedness.
