In this study, we demonstrate the use of physics-informed neural networks (PINNs) for computing the quasinormal modes (QNMs) of black holes in de Rham-Gabadadze-Tolley (dRGT) massive gravity. These modes describe the oscillation frequencies of perturbed black holes and are important in understanding the behavior of these objects. We show that by carefully selecting the hyperparameters of the PINN, including the network architecture and the training data, it is possible to achieve good agreement between the computed QNMs and the approximate analytical formula in the near-extremal limit for the smallest mode number. Our results demonstrate the effectiveness of PINNs for solving inverse problems in the context of QNMs and highlight the potential of these algorithms for providing valuable insights into the behavior of black holes.