Abstract: The macroscopic mechanical properties of composite structures are closely related to their mesoscopic architectures. Concurrent multiscale methods establish a direct link between macroscopic structural responses and mesoscopic material behaviors, providing high prediction accuracy. However, these methods require repeated solutions of the mesoscopic model at each integration point of the macroscopic structure, resulting in high computational costs and limited efficiency. To address this issue, this study develops a FEM×FFT concurrent multiscale computational framework based on the Fast Fourier Transform (FFT). A modified discrete Green operator is introduced into the mesoscopic FFT homogenization analysis to improve the accuracy of local stress field prediction. A unidirectional boron fiber-reinforced aluminum matrix composite is considered, and a mesoscopic representative volume element (RVE) is constructed and discretized using voxel-based grids. The FFT-based homogenization analysis is performed by solving the Lippmann–Schwinger equation, and the proposed framework is further applied to concurrent multiscale simulations of composite structures. The mesoscopic RVE results demonstrate that the macroscopic effective stress responses predicted by the FFT method agree well with those obtained from the finite element method, with a relative error below 7.4%. Furthermore, the predicted local stress distributions at the mesoscopic scale show excellent agreement with finite element results. Compared with the conventional FEM×FVDAM multiscale approach, the proposed FEM×FFT method accurately predicts the structural response, while reducing the computational time by approximately 78% and decreasing memory consumption by approximately 93%. In summary, the FEM×FFT concurrent multiscale framework provides an efficient and accurate approach for the multiscale mechanical analysis of complex composite structures.