Traditional control methods effectively manage robot operations using models like motion equations but face challenges with issues of contact and friction, leading to unstable and imprecise controllers that often require manual tweaking. Reinforcement learning, however, has developed as a capable solution for developing robust robot controllers that excel in handling contact-related challenges. In this work, we introduce a deep reinforcement learning approach to tackle the soft-capture phase for free-floating moving targets, mainly space debris, amidst noisy data. Our findings underscore the crucial role of tactile sensors, even during the soft-capturing phase. By employing deep reinforcement learning, we eliminate the need for manual feature design, simplifying the problem and allowing the robot to learn soft-capture strategies through trial and error. To facilitate effective learning of the approach phase, we have crafted a specialized reward function that offers clear and insightful feedback to the agent. Our method is trained entirely within the simulation environment, eliminating the need for direct demonstrations or prior knowledge of the task. The developed control policy shows promising results, highlighting the necessity of using tactile sensor information. The code and simulation results are available at Soft_Capture_Tactile repo.