Laser-driven neutron sources could offer a promising alternative to those based on conventional accelerator technologies in delivering compact beams of high brightness and short duration. We examine this through particle-in-cell and Monte Carlo simulations, that model, respectively, the laser acceleration of protons from thin-foil targets and their subsequent conversion into neutrons in secondary lead targets. Laser parameters relevant to the 0.5 petawatt (PW) LMJ-PETAL and 0.6-6 PW Apollon systems are considered. Due to its high intensity, the 20-fs-duration 0.6 PW Apollon laser is expected to accelerate protons up to above 100 MeV, thereby unlocking efficient neutron generation via spallation reactions. As a result, despite a 30-fold lower pulse energy than the LMJ-PETAL laser, the 0.6 PW Apollon laser should perform comparably well both in terms of neutron yield and flux. Notably, we predict that very compact neutron sources, of ~ 10 ps duration and ~ 100 µm spot size, can be released provided the lead convertor target is thin enough (~ 100 µm). These sources are characterized by extreme fluxes, of the order of 10 23 n cm-2 s-1 , and even ten times higher when using the 6 PW Apollon laser. Such values surpass those currently achievable at large-scale accelerator-based neutron sources (~ 10 16 n cm-2 s-1), or reported from previous laser experiments using low-Z converters (~ 10 18 n cm-2 s-1). By showing that such laser systems can produce neutron pulses significantly brighter than existing sources, our findings open a path towards attractive novel applications, such as flash neutron radiography or laboratory studies of heavy-ion nucleosynthesis.