Summary: In this paper we consider a general continuous-state nonlinear branching process which can be identified as a nonnegative solution to a nonlinear version of the stochastic differential equation driven by Brownian motion and Poisson random measure. Intuitively, this process is a branching process with population-size-dependent branching rates and with competition. We construct a sequence of discrete-state nonlinear branching processes and prove that it converges weakly to the continuous-state nonlinear branching process by using tightness arguments and convergence criteria on infinite-dimensional space.