defined in a convex smooth and bounded domain 9b8aa2e84d3b676fa1f52f380e355c9" title="Click to view the MathML source">Ω of 8d9616131a586987a7b7a1fa36d23816" title="Click to view the MathML source">R3, with e8ca160730a92ce3f3abbaeb1b6" title="Click to view the MathML source">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 e86c8bf2e" title="Click to view the MathML source">s≥0, with bd3ec4d2c719278" title="Click to view the MathML source">a≥0, 94658f8ae9ecabc7c0a" title="Click to view the MathML source">b>0 and α>1. In line with Viglialoro (2016), where for 8d322462a162e145d2084b404ce802"> the global existence of very weak solutions bdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) to the system is shown for any nonnegative initial data e5852bdf679fc1701418596e597a6f7"> and under zero-flux boundary condition on a18152c972549dfe65472f47343f9172" title="Click to view the MathML source">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 bd049fb9f2b37691502"> the initial data are such that ‖u0‖Lp(Ω) and ‖∇v0‖L4(Ω) are small enough, then bdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) is uniformly-in-time bounded.