defined in a convex smooth and bounded domain e84d3b676fa1f52f380e355c9" title="Click to view the MathML source">Ω of R3, with e8ca160730a92ce3f3abbaeb1b6" title="Click to view the MathML source">χ>0 and endowed with homogeneous Neumann boundary conditions. The source b9e720dadf9275c9de4ea201" title="Click to view the MathML 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 a≥0, b>0 and α>1. In line with Viglialoro (2016), where for e802"> the global existence of very weak solutions e8e3e2b953f3867c909" 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 a61fc0df50ec1"> does not exceed a certain value and for b9f2b37691502"> the initial data are such that a0bfc00b53884d83624a329342d4" title="Click to view the MathML source">‖u0‖Lp(Ω) and e487fd9a712c0ac38223e97025" title="Click to view the MathML source">‖∇v0‖L4(Ω) are small enough, then e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) is uniformly-in-time bounded.