Constraint programming has gained prominence as an effective and declarative paradigm for modeling and solving complex combinatorial problems. Techniques based on local search have proved practical to solve real-world problems, providing a good compromise between optimality and efficiency. In spite of the natural presence of concurrency, there has been relatively limited effort to use novel massively parallel architectures, such as those found in modern Graphical Processing Units (GPUs), to speedup local search techniques in constraint programming. This paper describes a novel framework which exploits parallelism from a popular local search method (the Large Neighborhood Search method), using GPUs.