We show the existence of Nash equilibria for the Navier-Stokes system and for the Oseen system. We consider the cases of the Dirichlet boundary conditions and of the Navier slip boundary conditions. Then, we study the asymptotic behavior of the Nash equilibria as the friction coefficient goes to $\infty$ in the Navier slip boundary condition. More precisely, we show that the Nash equilibria for the Navier slip boundary condition converge towards a Nash equilibrium for the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.