As Houthi militants in Yemen attacked ships with missiles and drones in the Red Sea early last year, top U.S. officials gathered at the Pentagon to prepare military options for President Donald Trump. Their meetings repeatedly grew tense.

On one side were Gen. Michael “Erik” Kurilla, then overseeing U.S. operations in the Middle East, and other hawks ready for action. The general was keen on military action not only against the Houthis but also against Iran, which had backed the Yemeni paramilitary group for years with weapons and training, according to people familiar with the matter. Like others, they spoke on the condition of anonymity to discuss internal government deliberations.

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On the other side were...