Queueing networks are of interest for analyzing congestion and delay in communication networks and complex manufacturing systems. Many of these systems have stations(servers) that can process more than one class of customers or jobs, and have Markov feedback structure(we call this kind systems as multiclass queueing networks). Heavy traffic is a very common phenomenon in the queueing networks. In general, the exact analysis of such system is quite difficult to obtain, both theoretically and computationally. One way to cope with these difficulties is to develop methods of the diffusion approximations. Usually, before establishing the diffusion approximations, we need to obtain the fluid approximation. This motivates us to study the relationship between fluid approximations and diffusion approximations for multiclass queueing networks. In particular, we focus on priority multiclass queueing networks. For balanced queueing networks with priority service discipline, we relate the asymptotically stable with the equilibrium point, and examine the relation between the asymptotically stable and the diffusion approximation. Based on this relation, the sufficient condition for the existence of diffusion approximation is given in terms of fluid approximations.