This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function χ(v) and the growth term f(u) under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that 0 < χ(v) ≤ χ0/vk (k ≥ 1, χ0 > 0) and λ1 − µ1u ≤ f(u) ≤ λ2 − µ2u (λ1, λ2, µ1, µ2 > 0). It is shown that if χ0 is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota.