This paper deals with multilevel domain decomposition techniques inside a multigrid framework to solve fluidstructure interaction (FSI) problems via the Finite Element Method (FEM). The mathematical problem consists of the Navier-Stokes equations in the Arbitrary-Lagrangian-Eulerian (ALE) formulation coupled with a non-linear incompressible structure model (Neo-Hookean). The coupling between the structure and the fluid, due to the important added-mass effect, is enforced within a unique solver. This kind of problem can be tackled efficiently by domain decomposition techniques. The solution strategy presented in this work is to solve several small local subproblems over subdomain patches using Vanka smoothing inside a multigrid algorithm. The results of a benchmark are presented in order to show the potentiality of the proposed solution method in terms of accuracy and efficiency.