defined in a convex smooth and bounded domain Ω of R3, with χ>0 and endowed with homogeneous Neumann boundary conditions. The source g behaves similarly to the logistic function and verifies g(s)≤a−bsα, for s≥0, with a≥0, b>0 and α>1. In line with Viglialoro (2016), where for the global existence of very weak solutions beb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) to the system is shown for any nonnegative initial data and under zero-flux boundary condition on v0, we prove that no chemotactic collapse for these solutions may present over time. More precisely, we establish that if the ratio does not exceed a certain value and for the initial data are such that ‖u0‖Lp(Ω) and ‖∇v0‖L4(Ω) are small enough, then beb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) is uniformly-in-time bounded.