A membrane module for an immersed operation has a fiber bundle of hollow fiber membranes which are cast into a headpiece and in immersed operation are surrounded by a liquid to be filtered. The module has a permeate collection chamber connected to the headpiece with an outlet for permeate draining from the interior of the hollow fiber membranes, and a gas supply having a pipe guided by the headpiece. The pipe terminates in the interior of the fiber bundle and has a gas outlet for a gaseous medium, which after the transfer from the pipe to the liquid to be filtered rises as bubbles between the hollow fiber membranes. The fiber bundle is divided into sections and free spaces remain between the sections which extend from the pipe to the outer circumference of the headpiece and promote an inflow of the liquid into a foot area adjoining the headpiece.