Abstract This paper deals with a Neumann boundary value problem in a d-dimensional box Td=(0,π)d $\mathbb{T}^{d}=(0,\pi)^{d}$ ( d=1,2,3 $d=1, 2, 3$) for a nonlinear diffusion chemotaxis model with logistic source. By using the embedding theorem, the higher-order energy estimates and bootstrap arguments, the condition of chemotaxis-driven instability and the nonlinear evolution near an unstable positive constant equilibrium for this chemotaxis model are proved. Our result provides a quantitative characterization for early spatial pattern formation on the positive constant equilibrium. Finally, numerical simulations are carried out to support our theoretical nonlinear instability results.