In distributed storage systems, both the repair bandwidth and locality are important repair cost metrics to evaluate the performance of a storage code. Recently, Guruswami and Wooters proposed an optimal linear repair scheme based on Reed-Solomon codes for a single failure, improved the bandwidth of the classical repair scheme. In this paper, we consider the repair bandwidth of Generalized Reed-Muller (GRM) codes, which have good locality property. We generalize Guruswami and Wooters' repairing scheme to GRM codes for single failure, which has nontrivial bandwidth closing to the lower bound when the subfield is small. We further extend the repair scheme for multiple failures in distributed and centralized repair models, and compute the expectation of bandwidth by considering different erasure-patterns.