ERBIL, Iraq—The anti-Islamic State strategy the U.S. is developing first began to take shape a month ago after a series of increasingly urgent phone calls from this Kurdish city.

In one, Kurdish leader Masoud Barzani told Vice President Joe Biden the jihadists were within 25 miles of Erbil, which is both the capital of the Kurdish region and home to U.S. military, intelligence, diplomatic and corporate offices. The message: Unless the U.S. stepped in, Erbil could fall in days.

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