We present an analysis for the process of fluid infiltration into a deformable, thin and permeable web that moves in close proximity over a rigid and stationary solid cylinder. While this is a process of significant interest in a range of coating, printing and composites pultrusion processes, its hydrodynamics have received limited attention in the open literature. The flow in the film separating the web from the cylinder is described by lubrication theory, while fluid transfer into the web is governed by Darcy's law. The deformation of the web at each position is a linear function of the local gap pressure; this is consistent with the assumption of a thin and rigidly supported web. Our results indicate that the web/fluid interface is forced away from the cylinder surface as it approaches it and bounces back downstream from the minimum separation point. This behavior produces a non-symmetric gap between the adjacent surfaces, and this is shown to have critical influence on the final amount of penetrating fluid. The extent of fluid penetration is also found to be affected by the web elasticity (expressed by the dimensionless Ne number) and permeability (expressed in dimensionless form via (K) over cap) where under a specific Ne and (K) over cap combination a maximum penetration depth is obtained. Finally, we derive a closed-form asymptotic solution for the final infiltration depth in the limit of Ne<< 1 and (K) over cap << 1 and test its predictions against the above-mentioned numerical results.