In 2015, a military coalition led by Saudi Arabia began bombing the Houthis, who have responded by firing missiles into the kingdom. Yet there is no evidence that the Houthis directly threaten the United States; they are an unsophisticated militant group with no operations outside Yemen and have not been classified by the American government as a terrorist group.

The Green Berets, the Army’s Special Forces, deployed to the border in December, weeks after a ballistic missile fired from Yemen sailed close to Riyadh, the Saudi capital. The Saudi military said it intercepted the missile over the city’s international airport — a claim that was cast in doubt by an analysis of photos and videos of the strike. But it was enough for Crown Prince Mohammed bin Salman to renew a longstanding request that the United States send troops to help the kingdom combat the Houthi threat.

A half-dozen officials — from the United States military, the Trump administration, and European and Arab nations — said the American commandos are training Saudi ground troops to secure their border. They also are working closely with American intelligence analysts in Najran, a city in southern Saudi Arabia that has been repeatedly attacked with rockets, to help locate Houthi missile sites within Yemen.

Along the porous border, the Americans are working with surveillance planes that can gather electronic signals to track the Houthi weapons and their launch sites, according to the officials, all of whom spoke on condition of anonymity because they were not authorized to discuss the mission publicly.

During a meeting on Capitol Hill in March, senators pressed Pentagon officials about the military’s role in the Saudi-led conflict, demanding to know whether American troops were at risk of entering into hostilities against the Houthis.

Pentagon officials told the senators what had already been said publicly: that American forces stationed in Saudi Arabia only advised within the kingdom’s borders and were focused mostly on border defense.

“We are authorized to help the Saudis defend their border,” Gen. Joseph L. Votel, the head of United States Central Command, told the Senate Armed Services Committee on March 13. “We are doing that through intelligence sharing, through logistics support and through military advice that we provide to them.”