A large variety of communication systems, including telephone and data networks, can be represented by so-called Whittle networks. The stationary distribution of these networks is insensitive, depending on the service requirements at each node through their mean only. These models are of considerable practical interest as they lead to engineering rules which are robust to the evolution of traffic characteristics. We consider the problem of optimal dynamic load balancing in multi-class stochastic networks, which is known to be a difficult optimization problem. By restricting the admissible set of solutions to the solutions preserving the structure of Whittle networks, we investigate the possibility of obtaining useful and robust approximations of the optimal performance.