This paper considers the Cauchy problem to the following multidimensional nonlinear system: \[ \begin{cases} \partial_t u-\Delta u =\operatorname{div}(uv), \quad (t,x)\in\mathbb{R}^+\times \mathbb{R}^d,\\ \partial_t v-\nabla u =0,\\ (u,v)|_{t=0}=(u_0,v_0), \end{cases} \] The main contribution of this paper is to study the well-posedness and ill-posedness of the above system in critical Besov spaces \(\dot{B}_{p,1}^{\frac{d}{p}-2}(\mathbb{R}^d)\times\Big(\dot{B}_{p,1}^{\frac{d}{p}-2}(\mathbb{R}^d)\Big)^d (d\geq 2)\). More precisely, the authors show that for \(1\leq p<2d\), then the locally well-posedness for large initial data and globally well-posedness for small initial data of the above system are proved. The present paper is devoted to establishing the ill-posedness in the sense that a “norm inflation” phenomenon occurs for \(p>2d\), that is, some arbitrarily small initial data can produce solutions arbitrarily large in critical Besov spaces after an arbitrarily short time. Here the ill-posedness result of the multidimensional Keller-Segel type chemotaxis system are new. The proofs of main results are delicate and seem to be right. The paper is well organized with a complete list of relevant references. The presentation of the paper is very clear. I think it is a good paper. But the question that for the large initial data whether the system is globally well-posed in the critical Besov Space is still open. This will inspire more researchers to study the system extensively.