A stopping criterion for fast Fourier transform (FFT)-based iterative schemes in computational homogenization is proposed and investigated numerically. This criterion is based on the separate evaluation and comparison of the discretization and iteration errors on the computed fields. Some estimators for these errors are proposed and their performances are assessed on a set of 2D problems in the frameworks of both the classical FFT-based methods and these that use a modified version of the featured Green's operator. In particular, two novel strategies for estimating the discretization error are investigated: either using an image processing approach or transposing to the FFT-based setting the constitutive relation error that is well-established in the context of the finite element method. It is then shown that the resulting stopping criterion leads to a better control of the global error on the computed effective property compared to the classical criterion based on the residual of the iterative scheme alone.