A better understanding of physical process of the fluid-seabed-structure interaction (FSSI) is beneficial for engineers involved in the design of marine infrastructures. Most previous studies for the problem of FSSI have considered wave-only conditions, despite the co-existence of wave and current in the real ocean environment. Unlike the previous studies, currents are included in the present study for the numerical modelling of FSSI, using an integrated FVM-FEM scheme, in which the VARANS equation is used to simulate fluid field, while Biot's poro-elastic model is used for porous flow in a seabed. Numerical examples show the important influences of currents on the local hydrodynamic process and the resulting dynamics of seabed foundation around a submerged rubble mound breakwater. The structure is relatively stable in the presence of counter-current waves, whereas the co-current waves would significantly compromise the instability of structure due to potential of shear failure and liquefaction in its sandy seabed foundation.