The monotone variational inequalities are being widely used as mathematical tools for studying optimal control problems and convex programming. In this paper, we propose a new prediction-correction method for monotone variational inequalities with linear constraints. The method consists of two procedures. The first procedure (prediction) utilizes projections to generate a predictor. The second procedure (correction) produces the new iteration via some minor computations. The main advantage of the method is that its main computational effort only depends on evaluating the resolvent mapping of the monotone operator, and its primal and dual step sizes can be enlarged. We prove the global convergence of the method. Numerical results are provided to demonstrate the efficiency of the method.