Synthesizing QA pairs via question generator (QG) for data augmentation is widely used in Machine Reading Comprehension (MRC), especially in data-scarce scenarios like limited labeled data or domain adaptation. However, the quality of generated QA pairs varies, and it is necessary to select the ones with high quality from them. Existing approaches focus on downstream metrics to choose QA pairs, which lacks generalization across different metrics and datasets. In this paper, we propose a general selection method that employs a generative large pre-trained language model as a reward model in a Reinforcement Learning (RL) framework for the training of the selection agent. Our experiments on both generative and extractive datasets demonstrate that our selection method leads to better downstream performance. We also find that using the large language model (LLM) as a reward model is more beneficial than using it as a direct selector or QA model. Furthermore, we assess the selected QA pairs from multiple angles, not just downstream metrics, highlighting their superior quality compared to other methods. Our work has better flexibility across metrics, provides interpretability for the selected data, and expands the potential of leveraging generative large language models in the field of MRC and RL training. Our code is available at https://github.com/JulieJin-km/LLM_RL_Selection.