In this paper, we study singular integrals on product domains with kernels in $L(\log L)^2(\mathbf{S^{n-1}}\times\mathbf{S^{m-1}})$ supported by surfaces of revolutions. We prove that our operators are bounded on $L^p$ under certain convexity assumption on the surfaces. Also, in this paper we prove that the convexity assumption is not necessary for the $L^2$ boundedness to hold. Moreover, additional related results are presented. Our condition on the kernel is known to be optimal.