Purpose: To improve rank constrained reconstructions for undersampled multi-image MRI acquisitions. Methods: Motivated by the recent developments in low-rank matrix completion theory and its applicability to rapid dynamic MRI, a new reordering-based rank constrained reconstruction of undersampled multi-image data that uses prior image information is proposed. Instead of directlyminimizing the nuclear norm of a matrix of estimated images, the nuclear norm of reordered matrix values is minimized. The reordering is based on the prior image estimates. The method is tested on brain diffusion imaging data and dynamic contrast enhanced myocardial perfusion data. Results: Good quality images from data undersampled by a factor of three for diffusion imaging and by a factor of 3.5 for dynamic cardiac perfusion imaging with respiratory motion were obtained. Reordering gave visually improved image quality over standard nuclear norm minimization reconstructions. Root mean squared errors with respect to ground truth images were improved by 18% and 16% with reordering for diffusion and perfusion applications, respectively. Conclusions: The reordered low-rank constraint is a way to inject prior image information that offers improvements over a standard low-rank constraint for undersampled multi-image MRI reconstructions. C 2015 American Association of Physicists in Medicine.