defined in a convex smooth and bounded domain Ω of R3, with baeb1b6" 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 e9dedaf5e86c8bf2e" title="Click to view the MathML source">s≥0, with a≥0, 8f8ae9ecabc7c0a" title="Click to view the MathML source">b>0 and 8f44717dfd1b7c521302687370734a" 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 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 a61fc0df50ec1"> does not exceed a certain value and for the initial data are such that 8faa0bfc00b53884d83624a329342d4" 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.