The Trump administration is considering closing the recently reopened United States Embassy in Havana after 21 Americans associated with the embassy experienced a host of unexplained health problems.

Secretary of State Rex W. Tillerson said during an appearance on CBS’s “Face the Nation” on Sunday that such a closing was “under evaluation.”

“It’s a very serious issue with respect to the harm that certain individuals have suffered,” he said. “We’ve brought some of those people home. It’s under review.”

The Trump administration has already reversed crucial pieces of what President Trump has called a “terrible and misguided deal” with Cuba that was struck during the Obama administration, but closing the embassy would be the most dramatic action yet to return the relationship to its Cold War deep freeze.

A closing of the embassy, were it to occur, would be less a political statement than one of concern over the risks that employees face in Havana. The American Foreign Service Association reported this month that the symptoms among those affected included mild traumatic brain injury, permanent hearing loss, loss of balance, severe headaches and brain swelling.

While noting that Cuba is responsible for protecting the health of diplomats posted to the country, State Department officials have yet to suggest that the Cuban government was behind the attacks. The Associated Press reported this weekend that the initial reaction by the Cuban president, Raúl Castro, to the news — apparent concern, with none of the usual how-dare-you-accuse-us attitude — had caught American officials off guard.

The Cubans even offered to let the F.B.I. go to Havana and investigate, a rare level of openness that suggested to some American officials that the Cuban government was equally baffled about the cause. Victims told The A.P. how they walked in and out of what seemed like powerful beams of sound that hit only certain rooms or even only parts of rooms.

American officials have speculated that the problems may have resulted from some sort of sonic attack or perhaps a surveillance operation gone wrong. The attack may have been the work of a rogue government unit or another government like Russia. That a Canadian diplomat was also affected deepened the mystery. Relations between Canada and Cuba have long been warm.

While the Trump administration has moved to reinstate travel and commercial restrictions on Cuba, there has appeared to be little appetite to entirely undo measures that are broadly popular, including among Republicans. That is another reason the administration has reacted cautiously.

Still, the attacks have led to growing concern on Capitol Hill. On Friday, five Republican senators sent a letter to Mr. Tillerson asking that he expel all Cuban diplomats in the United States and, if Cuba does not take tangible action, close the American Embassy in Havana.

American and Cuban officials met in Washington that day as part of a continuing law enforcement dialogue, and the subject of the attacks was raised, according to the State Department.

“It is an aggressive investigation that continues, and we will continue doing this until we find out who or what is responsible for this,” Heather Nauert, the department’s spokeswoman, said on Thursday.

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