  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.
