Studies on stress sensitivity of relative permeability in the laboratory were mostly carried out by varying overburden pressure at constant pore pressure (PP). However, in a real oilfield situation, changes in net stress are induced by changes in PP rather than overburden pressure. This research presents the effect of PP variation at a confining stress of 25 MPa on water–oil and silica nanofluid–oil relative permeability curves. Results showed that, at low PP variations of 0.1, 2, and 5 MPa, two-phase flow parameters exhibited distinct trends. Thus, for water–oil relative permeability, initial water (S_wi) and residual oil (S_or) saturations both decreased as PP increased. End-point oil K_ro (S_wi) and water K_rw (S_or) relative permeabilities both increased for water flooding. Similar trends were observed for nanofluid flooding. End-point mobility ratio and displacement efficiency both increased with increase in PP within this range for water and nanofluid flooding. At 10 MPa PP, S_wi decreased compared to low PP variations, while S_or increased significantly. Similar observation was made for nanofluid flooding except for S_or which had moderate increment. End-point mobility ratio increased significantly for water flooding, but had a slight increase for nanofluid flooding compared to low PP variations. Displacement efficiency decreased for both water and nanofluid flooding compared to low PP variations. At the same PP, water–oil and nanofluid–oil relative permeabilities showed similar S_wi and K_ro (S_wi) but differing S_or and K_rw (S_or). The findings provide insights into the effect PP variation on relative permeability and would be important in flooding design considerations.