Mr. Rapp came to the office with considerable experience: He served on an international tribunal on the atrocities in Rwanda and prosecuted war crimes in Sierra Leone before President Barack Obama appointed him to lead the office in 2009, making him the ambassador at large for war crimes.

Mr. Rapp also played a major role in arranging for one of Syria’s most important defectors to provide thousands of photographs to the F.B.I. of detainees who had perished in President Bashar al-Assad’s prisons.

In the two decades that it has existed, the office has played an important role in bringing perpetrators to justice. Mr. Rapp worked behind the scenes to press Kosovo to accept its new human rights tribunal. The office also pushed Senegal to try Hissène Habré, the former dictator of Chad.

But its fate has been caught up in the broader review that Mr. Tillerson has introduced to streamline decision-making at the State Department and by the Trump administration’s proposal to cut the agency’s budget by 31 percent.

The State Department spokeswoman said the agency was assessing “the utility of the nearly 70 special representatives or special envoys” who she said were often separated from the topical experts in the agency’s bureaus. The proposal to close the war crimes office was first reported by Foreign Policy.

Tom Malinowski, who served as the State Department’s top human rights official during the Obama administration, said it was important that the war crimes office and the person running it retained the stature to deal with international war crimes tribunals and senior foreign officials.

“They are not just advocating for accountability for war crimes; they are actively working to bring perpetrators to justice,” he said. “So you need somebody to run it who has the stature of a former prosecutor, a former judge or who is a recognized expert.”

“If all they do is take this office with its current resources, staff and ambassador and shift it elsewhere in the department, that’s not so big a problem,” he added. “But if the point is to downgrade it and strip it of resources, that would be a big setback.”