Quantum chaotic interacting N-particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales ∼logN. Here we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. Specifically, we consider quantum critical bosonic systems with attractive contact interaction that, despite being integrable, exhibit locally unstable dynamics in the corresponding many-body phase space of the large-N limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing ℏ/τ, again given by τ∼logN. This governs the long-time behavior of out-of-time-order correlators that feature quasi-periodic recurrences indicating reversibility.