The coupling through both drag force and volume fraction (of gas) of a kinetic equation of Vlasov type and a system of Euler or Navier-Stokes type (in which the volume fraction explicity appears) leads to the socalled thick sprays equations. Those equations are used to describe sprays (droplets or dust specks in a surrounding gas) in which the volume fraction of the disperse phase is non negligible. As for other multiphase flows systems, the issues related to the linear stability around homogeneous solutions is important for the applications. We show in this paper that this stability indeed holds for thick sprays equations, under physically reasonable assumptions. The analysis which is performed makes use of Lyapunov functionals for the linearized equations.