We introduce and study the notions of backward and forward I(α)-convergence and I-exhaustiveness of sequences of functions between asymmetric metric spaces. We establish a relation between backward (resp. forward) I(α)-convergent and backward (resp. forward) I-exhaustiveness. Also, we introduce and study ideal versions of some classical notions (Alexandroff and strong uniform) of convergence of sequences of functions in this context. We give some examples to ensure the alternation of basic results from the metric case.