This paper tackles the problem of how two selfish users jointly determine the operating point in the achievable rate region of a two-user Gaussian interference channel through bargaining. In previous work, incentive conditions for two users to cooperate using a simple version of Han-Kobayashi scheme was studied and the Nash bargaining solution (NBS) was used to obtain a fair operating point. Here a noncooperative bargaining game of alternating offers is adopted to model the bargaining process and rates resulting from the equilibrium outcome are analyzed. In particular, it is shown that the operating point resulting from the formulated bargaining game depends on the cost of delay in bargaining and how bargaining proceeds. If the associated bargaining problem is regular, a unique perfect equilibrium exists and lies on the individual rational efficient frontier of the achievable rate region. Besides, the equilibrium outcome approaches the NBS if the bargaining costs of both users are negligible.