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This article concerns the high-dimensional chemotaxis system with consumption of chemoattractant $$\displaylines{ u_t=\Delta u-\nabla\cdot(u\nabla v),\cr v_t=\Delta v-uv, }$$ under homogeneous boundary conditions of Neumann type, in a bounded domain $\Omega\subset\mathbb{R}^n~(n\geq 4)$ with smooth boundary. We prove that that if the initial data satisfy $u_0\in C^0(\overline{\Omega})$ and $v_0\in W^{1,q}(\Omega)$ for some $q>n$, this model possesses at least one global renormalized solution.