WASHINGTON—The U.S. military is providing its Gulf allies with intelligence to fine-tune their list of airstrike targets in Yemen’s most important port, one sign of the Trump administration’s deepening role in a looming assault that the United Nations says could trigger a massive humanitarian crisis.

While the U.N. is working furiously to broker a deal to avert a United Arab Emirates assault on a Red Sea port, the U.S. is helping the Gulf nation develop a list of targets meant to be off limits for airstrikes, American military...