While it is assumed in many applications that components are characterized by a homogeneous microstructure, this is not always the case. In fact, materials often exhibit heterogeneities, which can affect the material behavior drastically. Pronounced examples of this are Metal-Matrix Composites (MMCs). To determine the material behavior in multi-scale simulations, homogenization problems have to be solved that are often discretized using the Finite Element Method (FEM). The resulting algebraic system can then be solved by a (possibly non-linear) Conjugate Gradient (CG) scheme. However, it is a common problem that the system becomes ill-conditioned for finely resolved discretizations leading to slow solver convergence. Hence, preconditioners are used for CG, but these often have to be reassembled for any new microstructure. An exception are FFT-based preconditioners such as in Fourier-Accelerated Nodal Solvers (FANS, [1]) that are highly effective for specific homogenization problems with periodic boundary conditions. FANS act on a Fourier representation of the residual field and perform convolutions with a problem-specific fundamental solution. Recently, Fourier Neural Operators (FNOs, [2]) are emerging as a machine learning method to predict the solution of parametric PDEs. However, there is no guarantee that the predictions fulfill a given accuracy or are even physically consistent. A promising option is, thus, to use hybrid methods that combine the advantages of both data-driven methods and classical solvers. With FNO-CG, we propose a novel hybrid solver with guaranteed convergence involving special variants of FNOs that are admissible as preconditioners for CG and can be interpreted as a machine-learned extension of FANS. In contrast to FANS, FNO-CG is not restricted to periodic boundary conditions. For various problem formulations, we demonstrate that GPU-based implementations of FNO-CG are competitive with ML models in terms of performance but can achieve results with an accuracy that is multiple orders of magnitude higher. Thus, this represents a step towards realistic multiscale simulations in the spirit of FE² (FE Square).