ISTANBUL—Reformers and hard-liners in the Iranian regime have been jockeying for influence for decades. As the Islamic Republic now nears a landmark nuclear agreement with the U.S. and other world powers, the Middle East’s future depends on which force emerges as the deal’s main beneficiary.

Will the accord empower the more pragmatic factions interested in normalizing Iran’s relations with the West, even as the regime maintains repression at home? Or, as some of Iran’s neighbors fear, will the financial windfall from the lifting...