6D object pose estimation is widely applied in robotic tasks such as grasping and manipulation. Prior methods using RGB-only images are vulnerable to heavy occlusion and poor illumination, so it is important to complement them with depth information. However, existing methods using RGB-D data cannot adequately exploit consistent and complementary information between RGB and depth modalities. In this paper, we present a novel method to effectively consider the correlation within and across both modalities with attention mechanism to learn discriminative and compact multi-modal features. Then, effective fusion strategies for intra- and inter-correlation modules are explored to ensure efficient information flow between RGB and depth. To our best knowledge, this is the first work to explore effective intra- and inter-modality fusion in 6D pose estimation. The experimental results show that our method can achieve the state-of-the-art performance on LineMOD and YCBVideo dataset. We also demonstrate that the proposed method can benefit a real-world robot grasping task by providing accurate object pose estimation.