This article proposes a modulated model predictive control (MPC) with finite-control-set (FCS) to tackle the LC resonance problem in grid-connected current source converter (CSC). The proposed scheme pre-calculates the duty cycle of two active current vectors and the zero current vector for three virtual vectors, and then a cost function, in which the capacitor voltage feedback is introduced besides the grid current error penalization, is designed to select the optimized modulated vector. Therefore, the FCS reduces to only three virtual vectors during each predictive horizon with low computational burden. The proposed method not only realizes the optimization process including vectors selection and duty cycle determination simultaneously, but also can achieve superior steady-state performance and well dynamic response without LC resonance. Constant switching frequency can be realized via embedding space vector modulation (SVM) into the MPC algorithm. Capacitor voltage is estimated by an observer instead of using transducers to save hardware cost. Simulated and experimental results have verified the correctness and effectiveness of the proposed method.