defined in a convex smooth and bounded domain e84d3b676fa1f52f380e355c9" title="Click to view the MathML source">Ω of a586987a7b7a1fa36d23816" 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 895e9dedaf5e86c8bf2e" title="Click to view the MathML source">s≥0, with a5418049acc909fbd3ec4d2c719278" title="Click to view the MathML source">a≥0, b>0 and b7c521302687370734a" title="Click to view the MathML source">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 b7a61fc0df50ec1"> does not exceed a certain value and for 89" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816301225-si16.gif"> the initial data are such that ‖u0‖Lp(Ω) and ‖∇v0‖L4(Ω) are small enough, then e8e3e2b953f3867c909" title="Click to view the MathML source">(u,v) is uniformly-in-time bounded.