We study numerically the dynamical evolution of gas accumulated within 1kpc of nuclei in galaxy mergers. In particular, the effects of self- gravity of gas on gas transfer from 1kpc to 50 pc in the late phase of mergers are investigated. We find that, if the ratio of the gas mass to the mass of the two galactic cores is smaller than 0.2, the self-gravity of the gas is not a key determinant of gas dynamics in the central region of the merger. This is because the dynamical heating by two sinking cores is so strong. We also find that a large mass of gas (several 10^7 M_solar) can be efficiently transferred to the central 50 pc, where a supermassive black hole (the mass of which exceeds 10^8 M_solar) begins to dominate the gravitational potential, only if the cores of precursors are very compact (scalelength less than 10 pc) and the precursors initially have a large amount of gas (~10^9 M_solar) within the central 1kpc. Our numerical results predict that mergers between two late-type disc galaxies, both with compact cores, are promising candidates for quasars.