Autonomous navigation in offroad environments has been extensively studied in the robotics field. However, navigation in covert situations where an autonomous vehicle needs to remain hidden from outside observers remains an underexplored area. In this paper, we propose a novel Deep Reinforcement Learning (DRL) based algorithm, called CoverNav, for identifying covert and navigable trajectories with minimal cost in offroad terrains and jungle environments in the presence of observers. CoverNav focuses on unmanned ground vehicles seeking shelters and taking covers while safely navigating to a predefined destination. Our proposed DRL method computes a local cost map that helps distinguish which path will grant the maximal covertness while maintaining a low cost trajectory using an elevation map generated from 3D point cloud data, the robot's pose, and directed goal information. CoverNav helps robot agents to learn the low elevation terrain using a reward function while penalizing it proportionately when it experiences high elevation. If an observer is spotted, CoverNav enables the robot to select natural obstacles (e.g., rocks, houses, disabled vehicles, trees, etc.) and use them as shelters to hide behind. We evaluate CoverNav using the Unity simulation environment and show that it guarantees dynamically feasible velocities in the terrain when fed with an elevation map generated by another DRL based navigation algorithm. Additionally, we evaluate CoverNav's effectiveness in achieving a maximum goal distance of 12 meters and its success rate in different elevation scenarios with and without cover objects. We observe competitive performance comparable to state of the art (SOTA) methods without compromising accuracy.