Context. Adaptive optics (AO) is a technique allowing to drastically improve ground-based telescopes angular resolution. The wavefront sensor (WFS) is one of the key components of such systems, driving the fundamental performance limitations. Aims. In this paper, we focus on a specific class of WFS: the Fourier-filtering wavefront sensors (FFWFS). This class is known for its extremely high sensitivity. However, a clear and comprehensive noise propagation model for any kind of FFWFS is lacking. Methods. Considering read-out noise and photon noise, we derive a simple and comprehensive model allowing to understand how these noises propagates in the phase reconstruction in the linear framework. Results. This new noise propagation model works for any kind of FFWFS, and allows to revisit the fundamental sensitivity limit of these sensors. Furthermore, a new comparison between widely used FFWFS is held. We focus on the two main used FFWFS classes: the Zernike WFS (ZWFS) and the pyramid WFS (PWFS), bringing new understanding of their behavior.