Let X and Y be two Banach spaces. A bounded operator T : X −→ Y is said to be a BS-compact operator whenever T sends Banach-Saks subsets of X onto norm compact sets of Y ([20]). In this paper, our central focus is upon introducing the class of almost BS-compact operators. The paper rests essentially on two parts. The first is devoted to the connection of this new class of operators with classical notions of operators, such as BS-compact operators, AM-compact operators, and Dunfort-Pettis operators. The second part is dedicated to the domination problem within the framework of (almost) BS-compact operators. Mathematics Subject Classification. 46B42, 47B60, 47B65.