defined in a convex smooth and bounded domain Ω of 8d9616131a586987a7b7a1fa36d23816" title="Click to view the MathML source">R3, with aeb1b6" title="Click to view the MathML source">χ>0 and endowed with homogeneous Neumann boundary conditions. The source e720dadf9275c9de4ea201" title="Click to view the MathML source">g behaves similarly to the logistic function and verifies g(s)≤a−bsα, for s≥0, with 9acc909fbd3ec4d2c719278" title="Click to view the MathML source">a≥0, ae9ecabc7c0a" 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 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 e487fd9a712c0ac38223e97025" title="Click to view the MathML source">‖∇v0‖L4(Ω) are small enough, then bdf8dbeb63e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) is uniformly-in-time bounded.