In this paper we study the fully parabolic chemotaxis system with logistic-type source and nonlinear production: ut=Δu−χ∇⋅(u∇v)+f(u), vt=Δv−v+g(u), subject to the non-flux boundary conditions with a smooth and bounded domain Ω⊂Rn (n≥1), and the nonnegative initial data u0∈C0(Ω̄) and v0∈W1,∞(Ω), where the sensitivity χ>0, the logistic-type source f(s)≤s−μsα for s≥0 with α>1, μ>0, and the nonlinear production g(s)≤s(s+1)β−1 for s≥0 with β>0. It is obtained that the problem possesses a globally bounded classical solution if the self-restriction mechanism of the logistic-type source dominates the nonlinear production that β<α−1, or β=α−1 with μ>0 sufficiently large. This extends the current global existence result by Nakaguchi and Osaki (2018). The situation with the nonlinear production controlled by the linear diffusion that β∈(0,2n) is considered as well, which is parallel to the model with nonlinear diffusion and subcritical chemotactic sensitivity without logistic source studied by Tao and Winkler (2012).