Motion artifacts in Magnetic Resonance Imaging (MRI) are one of the frequently occurring artifacts due to patient movements during scanning. Motion is estimated to be present in approximately 30% of clinical MRI scans; however, motion has not been explicitly modeled within deep learning image reconstruction models. Deep learning (DL) algorithms have been demonstrated to be effective for both the image reconstruction task and the motion correction task, but the two tasks are considered separately. The image reconstruction task involves removing undersampling artifacts such as noise and aliasing artifacts, whereas motion correction involves removing artifacts including blurring, ghosting, and ringing. In this work, we propose a novel method to simultaneously accelerate imaging and correct motion. This is achieved by integrating a motion module into the deep learning-based MRI reconstruction process, enabling real-time detection and correction of motion. We model motion as a tightly integrated auxiliary layer in the deep learning model during training, making the deep learning model 'motion-informed'. During inference, image reconstruction is performed from undersampled raw k-space data using a trained motion-informed DL model. Experimental results demonstrate that the proposed motion-informed deep learning image reconstruction network outperformed the conventional image reconstruction network for motion-degraded MRI datasets.