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In this paper, we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators in two-dimensional setting in the following form: \begin{equation*} L_{\lambda }\left( f;x,y\right) =\underset{D}{\iint }f\left( t,s\right) K_{\lambda }\left( t-x,s-y\right) dsdt,\text{ }\left( x,y\right) \in D, \end{equation*} where $D=\left \langle a,b\right \rangle \times \left \langle c,d\right \rangle $ is an arbitrary closed, semi-closed or open rectangle in $\mathbb{R}^{2}$ and $% \lambda \in \Lambda ,$ $\Lambda $ is a set of non-negative indices with accumulation point $\lambda_{0}$. Also, we provide an example to support these theoretical results. In contrast to previous works, the kernel function $K_{\lambda }\left( t,s\right) $ does not have to be even, positive or 2$\pi -$periodic.