To study the operator inequalities, the notions of class A operators and quasi-class A operators are developed up to recently. In this paper, quasi-$A(n,{kappa})$ class operator for $n{geq}2$ and ${kappa}{geq}0$ is introduced as a new notion, which generalizes the quasi-class A operator. We obtain some structural properties of these operators. Also we characterize quasi-$A(n,{kappa})$ classes for n and ${kappa}$ via backward extension of weighted shift operators. Finally, we give a simple example of quasi-$A(n,{kappa})$ operators with two variables.