In 1999, \textit{Y. Ding} [J. Math. Anal. Appl. 232, No. 1, 222--228 (1999; Zbl 0920.42013)] studied the maximal function \(M_{\Omega}(f)(x,y)=\sup_{h\in U}| (T_{\Omega,h}f)(x,y)| \), where \(U\) is the set of all \(h\in L^2(R_{+}\times R_{+}, r^{-1}s^{-1}dr ds)\), \(\| h\| _{L^2(R^{+}\times R_{+}, r^{-1}s^{-1}dr ds)}\leq 1\). He obtained that if \(\Omega\in L(\log L)^2\), then \(M_{\Omega}\) is bounded on \(L^2\). In this paper, the author studies the \(L^p\)-boundedness of this maximal operators with rough kernels in \(L(\log L)\). He proves that the operators are bounded on \(L^p\) for \(2\leq p<\infty.\) Morever, he shows that the condition on the kernel is optimal in the sense that the space \(L(\log L)\) cannot be replaced by \(L (\log L)^r\) for any \(r<1\).