We propose a model-data asymptotic-preserving neural network (MD-APNN) method to solve the nonlinear gray radiative transfer equations (GRTEs). The system is challenging to be simulated with both the traditional numerical schemes and the vanilla physics-informed neural networks (PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving (AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the proposed method, and a number of numerical examples are presented to illustrate the efficiency of MD-APNNs, and particularly, the importance of the AP property in the neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure Data-driven networks in the simulation of the nonlinear non-stationary GRTEs.