Two cylinders rotating next to each other generate a large hydrodynamic force if the intermediate space is filled with a viscous fluid. Herein, we explore the case where the cylinders are separated by two layers of viscous immiscible fluids, in the limit of small capillary deformation of the fluid interface. As the interface deformation breaks the system's symmetry, a novel force characteristic of soft lubrication is generated. We calculate this capillary-lubrication force, which is split into velocity-dependant and acceleration-dependant contributions. Furthermore, we analyze the variations induced by modifying the viscosity ratio between the two fluid layers, their thickness ratio, and the Bond number. Unlike standard elastic cases, where a repelling soft-lubrication lift force has been abundantly reported, the current fluid bilayer setting can also exhibit an attractive force due to the non-monotonic deflection of the fluid interface when varying the sublayer thickness. Besides, at high Bond numbers, the system's response becomes analogous to the one of a Winkler-like substrate with a viscous flow inside.