In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $ℐ$-quasinormal convergence. We then introduce the notion of $ℐQN\left(ℐwQN\right)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $ℐ$-quasinormally convergent to $0$ (has a subsequence which is $ℐ$-quasinormally convergent to $0$) and make certain observations on those spaces.