Conventional turbulence models have a single time scale of turbulence which is determined by turbulent energy k and its dissipation rate ε (i.e., k/ε). This fact implicitly assumes that the shape of the turbulent energy spectrum is universal. Therefore, they cannot predict the flows that are influenced by a lot of time scales of turbulence. In simulating such flows it is neccessary to introduce some information about scales of turbulence. In order to overcome this problem, multiple-time-scale turbulence models have been proposed. However, most of them are based on the k-ε model and the eddy viscosity hypothesis restricts the universality. In the present study the basic form of a multiple-time-scale Reynolds stress model was proposed, based on the transport equation for a two-point velocity correlation. In addition, this model was applied to a grid turbulent flow. By introducing some restrictions and assumptions, the model coefficients were determined analytically and numerically. It was shown that the model is satisfactory.