defined in a convex smooth and bounded domain Ω of 87a7b7a1fa36d23816" title="Click to view the MathML source">R3, with χ>0 and endowed with homogeneous Neumann boundary conditions. The source bfb45a44b9e720dadf9275c9de4ea201" title="Click to view the MathML source">g behaves similarly to the logistic function and verifies 9f806d5d12fe346fe1cdd082c7124d4" title="Click to view the MathML source">g(s)≤a−bsα, for bf123895e9dedaf5e86c8bf2e" title="Click to view the MathML source">s≥0, with 9fbd3ec4d2c719278" title="Click to view the MathML source">a≥0, e9ecabc7c0a" title="Click to view the MathML source">b>0 and 87370734a" title="Click to view the MathML source">α>1. In line with Viglialoro (2016), where for the global existence of very weak solutions (u,v) to the system is shown for any nonnegative initial data 9fc1701418596e597a6f7"> and under zero-flux boundary condition on 9172" 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 87b32702defd0b7a61fc0df50ec1"> does not exceed a certain value and for bf05bbd049fb9f2b37691502"> the initial data are such that bfc00b53884d83624a329342d4" title="Click to view the MathML source">‖u0‖Lp(Ω) and e9e487fd9a712c0ac38223e97025" title="Click to view the MathML source">‖∇v0‖L4(Ω) are small enough, then (u,v) is uniformly-in-time bounded.