In recent years, non-cooperative objects, such as failed satellites and space junk, can be detected in space. These objects are usually manipulated or collected by free-floating dual-arm space manipulators. Reinforcement learning methods show more promise in trajectory planning for space manipulators as difficulties in modeling and manual parameter tuning have been recently surmounted. Although previous studies demonstrated their effectiveness, they cannot be applied to track dynamic targets with unknown rotation (non-cooperative objects). In this paper, we proposed a learning system for free-floating dual-arm space manipulator motion planning against non-cooperative objects. Specifically, our method consists of two modules. Module I realizes the multi-target trajectory planning for two end-effectors within a large target space. Next, Module II takes the point clouds of the non-cooperative object as input to estimate the motional properties, and then the location of the target points on the non-cooperative object can be predicted. Target points on rotating objects with unknown rotation can be successfully tracked by the end-effectors through the combination of Module I and Module II. Furthermore, experiments also demonstrate the scalability and generalization of our learning system.