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In a smoothly bounded domain $\Omega \subset \mathbb{R}^n$ , $n\ge 1$ , this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system \begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*} with parameter $\alpha \gt 0$ and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics.