The Obama administration and Iran, engaged in direct nuclear negotiations and facing a common threat from Islamic State militants, have moved into an effective state of détente over the past year, according to senior U.S. and Arab officials.

The shift could drastically alter the balance of power in the region, and risks alienating key U.S. allies such as Saudi Arabia and United Arab Emirates who are central to the coalition fighting Islamic State. Sunni Arab leaders view the threat posed by Shiite Iran as equal to or greater...