Clinical diagnosis and quantification of liver disease have been improved through the development of techniques using contrast-enhanced liver MRI sequences. To qualitatively or quantitatively analyze such image sequences, one first needs to correct for rigid and non-rigid motion of the liver. For motion correction of the liver, we have employed bi-directional local correlation coefficient Demons, which is a variation of the original Demons method. However, despite the intrinsic speed of the Demons method, the run-time on the order of an hour of its CPU-based implementation is not sufficiently short for a regular clinical use. For this reason we implemented the method on a graphics processing unit (GPU) using OpenCL. On NVIDIA GTX 260M, which is a laptop GPU, we achieved sub-minute runtime for the motion correction of typical liver MRI scans, which was 50 times faster than its CPU-based implementation. A sub-minute runtime of liver MRI motion correction allows for its regular clinical use.