In this paper, an interference-aware linear precoding scheme, called generalized singular value decomposition based interference suppression precoding (GSVD-ISP), is proposed. GSVD-ISP scheme employs a whitening filter for interference suppression at the receiver and a novel precoder considering other cells' interference for each user at the transmitter. Through slightly loosening the constraint on the signal-to-leakage-plus-noise (SLNR) maximization for multi-user multiple-input multiple-output (MU-MIMO) systems with multiple data streams per user and the use of GSVD algorithms, the proposed scheme decreases redundant computational complexity and reduces the gap between any couple of the stream effective channel gains compared with the original interference-aware enhancement to SLNR precoding scheme-multiuser generalized eigenmode transmission (MGET). Simulation results demonstrate that the proposed GSVD-ISP technique achieves considerable gains in terms of bit error rate (BER) over MGET technique with multi-stream and the traditional interference-aware block diagonalization (BD) scheme while maintaining roughly the same achievable sum-rate.