In recent years, the use of neural operators to precondition linear systems resulting from discretised partial differential equations has seen growing interest. However, the chosen architecture of the neural operator often restricts the linear solver to specific "flexible" Krylov methods or simple fixed-point iterations. Simultaneously, approximating solution operators to linear PDEs motivates a connection to Green's functions, which have been successfully applied to construct preconditioners, both with and without the use of machine learning. The Neural Green's Operator (NGO) is a neural operator architecture that is designed to approximate Green's functions for a parametric family of linear PDEs. NGOs have shown to be promising both as a neural operator for linear PDEs and for preconditioning such PDEs. In this work, NGOs are shown to be a natural choice as a data-driven coarse solver inside a two-level preconditioning strategy. When solving a PDE with a two-level preconditioner, each iteration of the linear solver involves at least one solution of a smaller ("coarse") linear system. Once trained, an NGO can accelerate this process for a whole class of linear PDEs by replacing the coarse solves by a computationally efficient and parallelisable neural network. The NGO architecture is particularly suitable for this application, as a trained operator can be used as a coarse solver with minimal modifications. NGO-based two-level preconditioners are found to result in similar convergence rates to classical two-level schemes, while being potentially much more computationally efficient.