In this paper, we define the $ij$-almost-set Menger ( $ij$-ASM) property in bitopological spaces. We put up some equivalences of $ij$-almost-set Menger bitopological spaces and investigate the behaviours of such spaces under some different types of mappings. We later take the preservation of these properties under union, subspaces, products into consideration and give some related examples. We finally introduce the concept of $ij$-almost $P_{\gamma }$-set in bitopological spaces.

This paper is a continuation and complement for previous works on selective covering properties. We introduce the novel concept of the nearly Menger property in a bitopological context. We demonstrate its interrelations with existing covering properties and construct certain equivalences between those. We also investigate various properties of nearly Menger bitopological spaces by considering it under subspaces, products, and certain type of mappings.