Spontaneous four-wave mixing originated from third-order (Kerr) nonlinear effect in fiber, is an efficient way to produce quantum correlated photon pairs. If the commercially available high-nonlinear fiber can be used as the Kerr nonlinear medium, it will be beneficial for simplifying the experimental setup, because the length of fiber and power of pump can be reduced. Near-degenerate photon pairs in telecom band are produced in 20 m-long high nonlinear fiber by pulsed laser. The influence of weak-birefringence of high nonlinear fiber upon the production efficiency of photon pairs is experimentally investigated and analyzed. When the polarization of pump is along the fast or slow axis of the high nonlinear fiber, the production efficiency of photon pairs is maximal, and photon pairs are co-polarized with the pump; when the polarization of pump is along the 45° in respect to the fast axis, the production efficiency of photon pairs is minimal, which is about 80% of maximum.