According to the definition of the orbital angular momentum (OAM) density, it can be deduced that the high-order Bessel beams have the OAM just as the high-order Laguerre-Gaussian beams and its divergence angle is zero. Based on this theory, an optical fiber of ring refractive index distribution is proposed to transmit the high-order Bessel OAM beams. A theoretical analysis of the OAM modes which can propagate in this optical fiber is presented and graphs of the intensity and phase distributions of these modes are obtained by simulation. The analysis result is that the number of the OAM modes transmitted in the fiber does not vary linearly with the change of any single parameter, and it depends on the number of all Bessel function roots between the normalized frequency range of the inner and outer diameter of the ring fiber.