We propose a method to couple Finite Volume method (FVM) and Finite Element method (FEM) for fluid-structure interaction (FSI) problems with different time-steps depending on the fluid and solid subdomains. In the context of inviscid compressible fluids, fluid-structure coupling methods are generally based on a partitioned approach. Compared to a monolithic approach, partitioned methods have a relatively low numerical cost and are simpler to implement. However, due to the time-lag associated with these methods, they can lack stability and accuracy. These problems are exacerbated when we wish to perform calculations with different time steps. The idea of the proposed method is to achieve a compromise between the advantages of both partitioned and monolithic methods. Fluid and structure equations are solved simultaneously, thus eliminating the time-lag concerning the data exchange. An important part of the calculation is carried out independently for fluid and structure subdomains, while ensuring the continuity of normal velocities between the fluid and structure thanks to the introduction of Lagrange multipliers. Furthermore, it enables us to perform explicit/implicit multi time step FSI computations with an improved stability. The method is based on a dual Schur method, called the GC method, which extends the FETI techniques. This method was originally proposed for structural dynamics and then extended to FSI using the SPH method for fluid and the FEM for structure. Our method adapts these approaches to the FVM to resolve fluid and to the FEM for structure. For fluid, explicit Runge Kutta schemes and implicit Backward Euler time integrators have been tested. Concerning the structure subdomain, implicit and explicit Newmark type schemes have been used. The method has been tested using various 1D and 2D cases, including the 1D piston case and the 2D floating case presented in the partitioned coupling work by Piperno and Farhat. The proposed coupling method turns out to be accurate and stable for these two benchmark fluid-structure interaction cases, without generating spurious energy at the interface as in partitioned approaches.