Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the nonparametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the $σ-$field generated by diffusion processes is destroyed, the classical central limit theorem for martingale difference sequences can not work. In this paper, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps. We believe that our proof for local polynomial threshold estimators provides a new method in this fields, especially local linear case.