In the framework of the circular restricted three-body problem (the Sun-planet-comet), we generalize the map of Liu &Sun (1994) to study the transfer of comets from near-parabolic to Halley-family ones, under the perturbations of Jupiter. Numerical esults show that the transfer is effective for comets on both the direct and retrograde orbits crossing the planet's orbit, and on the direct non-crossing orbits but with the perihelion distances close to the semi-major axis of the planet's orbit. The dependence of the transfer probability and of the average transfer time on the perihelion distances of the cometary orbits and on the planet mass are determined. The probabilities of comet transfer from the Oort Cloud are found to be 0:12 and 0:067 for comets on direct and retrograde orbits, respectively. This leads us to expect an average flux of 0:23 new Halley-family comets per year interior to Jupiter's orbit. The calculation shows that vabout86%of the flux comes from comets on the Jupiter-crossing orbits.