This study is dedicated to the rapid removal of protein aggregates and viruses from plasma-derived human serum albumin (HSA) product to reduce the risk of viral contamination and increase biosafety. A two-step filtration approach was implemented to first remove HSA aggregates and then achieve high model virus clearance using a nanocellulose-based filter paper of different thicknesses, i.e., 11 μm (prefilter) and 22 μm (virus filter) at pH 7.4 and room temperature. The pore size distribution of these filters was characterized by nitrogen gas sorption analysis. Dynamic light scattering (DLS) and size-exclusion high performance liquid chromatography (SE-HPLC) were performed to analyze the presence of HSA aggregates in process intermediates. The virus filter showed high clearance of a small-size model virus, i.e., log10 reduction value (LRV) > 5, when operated at 3 and 5 bar, but a distinct decrease in LRV was detected at 1 bar, i.e., LRV 2.65–3.75. The throughput of HSA was also dependent on applied transmembrane pressure as was seen by Vmax values of 110 ± 2.5 L m−2 and 63.6 ± 5.8 L m−2 at 3 bar and 5 bar, respectively. Protein loss was low, i.e., recovery > 90%. A distribution of pore sizes between 40 nm and 60 nm, which was present in the prefilter and absent in the virus filter, played a crucial part in removing the HSA aggregates and minimizing the risk of virus filter fouling. The presented results enable the application of virus removal nanofiltration of HSA in bioprocessing as an alternative to virus inactivation methods based, e.g., on heat treatment.