RIYADH—The nuclear deal that the U.S. and other world powers hope to reach with Iran would put a 10-year curb on the Islamic republic’s nuclear program. For some of Iran’s regional rivals, that is also becoming a deadline for developing nuclear arms of their own.

In Saudi Arabia, there are widespread public calls to match Iran’s nuclear quest. The two other Middle East heavyweights, Turkey and Egypt, could also feel compelled to follow suit, senior Western and Arab officials warn.

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