ABSTRACT Dwarf novæ (DNe) and low-mass X-ray binaries (LMXBs) are compact binaries showing variability on time-scales from years to less than seconds. Here, we focus on explaining part of the rapid fluctuations in DNe, following the framework of recent studies on the monthly eruptions of DNe that use a hybrid disc composed of an outer standard disc and an inner magnetized disc. We show that the ionization instability, which is responsible for the monthly eruptions of DNe, is also able to operate in the inner magnetized disc. Given the low density and the fast accretion time-scale of the inner magnetized disc, the ionization instability generates small, rapid heating and cooling fronts propagating back and forth in the inner disc. This leads to quasi-periodic oscillations (QPOs) with a period of the order of 1000 s. A strong prediction of our model is that these QPOs can only develop in quiescence or at the beginning/end of an outburst. We propose that these rapid fluctuations might explain a subclass of already observed QPOs in DNe as well as a, still to observe, subclass of QPOs in LMXBs. We also extrapolate to the possibility that the radiation pressure instability might be related to type B QPOs in LMXBs.