This paper is motivated by a new class of SDEs--PDEs systems, the so called Lagrangian stochastic models which are commonly used in the simulation of turbulent flows. We study a position--velocity system which is nonlinear in the sense of McKean. As the dynamics of the velocity depends on the conditional expectation w.r.t. its position, the interaction kernel is singular. We prove existence and uniqueness of the solution to the system by solving a nonlinear martingale problem and showing that the corresponding interacting particle system propagates chaos.