This paper deals with the chemotaxis system with nonlinear signal secretion \[ {\left\lbrace \begin{array}{ll} u_t=\nabla \cdot (D(u)\nabla u - S(u)\nabla v), &x\in \Omega ,\quad t>0,\\ v_t=\Delta v-v+g(u),&x\in \Omega ,\quad t>0, \end{array}\right.} \] under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n\ge 2$). The diffusion function $D(s) \in C^{2}([0,\infty ))$ and the chemotactic sensitivity function $S(s) \in C^{2}([0,\infty ))$ are given by $D(s) \ge C_{d} (1+s)^{-\alpha }$ and $0 < S(s) \le C_{s} s(1+s)^{\beta -1}$ for all $s\ge 0$ with $C_{d},C_{s}>0$ and $\alpha ,\beta \in \mathbb{R}$. The nonlinear signal secretion function $g(s) \in C^{1}([0,\infty ))$ is supposed to satisfy $g(s)\le C_{g} s^{\gamma } \text{ for all } s\ge 0$ with $C_{g},\gamma > 0$. Global boundedness of solution is established under the specific conditions: \[ 0