This paper deals with a modification of the classical Szász-Mirakjan type operators of two variables. It introduces a new sequence of non-polynomial linear operators which hold fixed the polynomials $x^{2}+\alpha x$ and $y^{2}+\beta y$ with $\alpha ,\beta \in [0,\infty)$ and we study the convergence properties of the new approximation process. Also, we compare it with Szász-Mirakjan type operators and show an improvement of the error of convergence in $[0,1] \times [0,1]$. Finally, we study statistical convergence of this modification.