WASHINGTON—Arab governments are privately expressing their concern to Washington about the emerging terms of a potential deal aimed at curbing Iran’s nuclear program, according to Arab and U.S. officials involved in the deliberations.

The direction of U.S. diplomacy with Tehran has added fuel to fears in some Arab states of a nuclear-arms race in the region, as well as reviving talk about possibly extending a U.S. nuclear umbrella to Middle East allies to counter any Iranian threat.

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