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Electrospun polymer membranes are regarded as prospective biosensor components due to their large specific surface area and diverse opportunities for chemical modifications. However, their intricate porous structure can impede diffusion and render some analyte-binding sites inaccessible. To overcome these diffusion limitations and improve analyte adsorption onto the polymer, a pressure-driven sample flow through the membrane can be employed. To date, the efficiency of pressure-driven analyte delivery into these membranes has not been quantified. Here, we compare forced flow and passive sample diffusion through poly(dioxanone) electrospun membranes. We examine two model analytes, BSA and interleukin-1 beta (IL1b), to address both non-specific and specific binding. Following exposure of the membranes to the test solutions, we measured the residual concentrations of the analytes using fluorometry and enzyme-linked immunosorbent assay (ELISA) techniques. The pressure-driven sample loading was superior to passive diffusion, with a 2.8–11.5-fold change for physical adsorption and a 2.4–3.4-fold difference for specific binding. Our data can be useful for the development of immunoassays and microfluidic devices.